Inverse matrix.

Engineering systems 24.09.2019
Engineering systems
Exactly a year ago, we tested the NEC MultiSync EA231WMI monitor - the first available model on the new E-IPS matrix. Despite the general impression, EA231WMI was still a niche monitor - relatively expensive and not equipped with a response compensation scheme, and therefore quite slow, it was well suited for people interested in the first place in high-quality color reproduction, but it is possible to call it universal and homemade It was only with reservations.

Since then, time has passed, the detachment of monitors on E-IPS matrices has been replenished with new fighters - largely due to the company Dell, who immediately released several models with very attractive characteristics and relatively low prices. Today we will look at them.

Testing technique

Description of the testing methodology used by us, as well as a brief explanation, which means in practice those or other passports or the parameters of monitors mean by us, can be found in the material " Methods of testing LCD monitors" If you feel that it is not enough good to focus in numbers and terms, which is replete with an article - please read the relevant sections of the specified description, we hope it will clarify many questions.

If you have not found the monitor you are interested in in this article, it makes sense to refer with full list Tested models.

Design and ergonomics

For Dell traditionally great importance It has been working for the needs of the corporate market, primarily American, distinguished by fair conservatism in matters of appearance - bright colors and brilliant surfaces by many people who choose equipment for work are regarded as "non-serious", not corresponding to the spirit of the business situation.

The same applies to people sensitive to color rendering: they are better that monitor that the least distracts attention from the image itself, which means that gloss, multilingual and supernatural LEDs they consider as unacceptable.

Is it worth it to be surprised in the world that the whole line of Dell monitors on IPS matrices calculated is obviously mainly of these two groups of people, is made in strict matte corners of dark gray, almost black? Of course, there is no: it should look like a high-quality professional monitor, which is chosen for its capabilities, quality and reliability, and not an appearance. However, how things are with opportunities and quality, we will look further, and now we will deal with external species and ergonomics, that is ease of use.



U2311h



U2711


At first glance, all five monitors externally differ only in size (in the pictures above for example, only two of them, believe me, the remaining three are the same): the rectangular body of dark gray matte plastic, the matte screen surface, stable rectangular stand and vertical Row of buttons to the right of the screen. The only decoration is the metallized inscription "Dell" in the center under the screen.



Rear monitors are just as ascetic, except that the silver stand adds a bit of a variety into the color gamut.



The stand provides all the functionality, which can only be adjusting the height (and monitors can drop almost to the table level), turn around the vertical axis (the base of the stand remains fixed), turning into portrait mode and adjusting the screen inclination. Moves and turns the monitor easily, without the application for significant efforts.



Buttons on u2211h


The control buttons in two younger models - U2211H and U2311H - mechanical, they are built into a vertical range of the right edge of the monitor and press with a light click. The bottom button is to turn off the power, the LED indicator is built into it, the LED indicator is inexplicit (when working) or yellow (in the "hibernation") color. Please note that there are no signatures from the buttons: the fact is that their functions are not specified hard, they can be changed from the monitor menu.



Buttons on U2410.


Senior models are slightly different: all buttons, except the power button, are made of sensory, and designated slightly protruding square heads of blue LEDs. When bringing a finger (even before touching the surface), all LEDs light up, so that you can easily get on the buttons even in complete darkness.

Some differences, besides the aesthetic, I did not find any differences between the sensory and mechanical buttons: both those and others are equally convenient and the same clearly work out. We look at the touch buttons with their sunbathing and extinguishing LEDs, of course, more advantageous, but on this their advantages ends.

A more significant difference between monitors from each other - the number of inputs and outputs.



U2211H and U2311H on the left side there is only a pair of USB connectors ...



... while three senior models are complemented by the SecureDigital Carton Girl. However, the usefulness is also not very high: for example, the author of this article, having on the monitor with side USB connectors and at home, and at work, and could not accustom themselves to use - uncomfortable every time either look for them to the touch or deploy the monitor to yourself sideways. Perhaps, you can put something permanently included there, for example, a wireless mouse receiver, but for flash drives, players and such equipment it is more convenient to use connectors on the system unit housing.



If we talk about the rear connectors, then the U2211H and U2311H are equipped with the inputs of DisplayPort, DVI and D-Sub, connectors for connecting the block purchased separately with speakers (it was not searched in Russia), one input and two more USB outputs in which it happens convenient Stick a mouse or keyboard.



Dell U2410 offers already a choice of two DVI, DisplayPort, HDMI, D-SUB, as well as component and composite video inputs. In the sound output, if you do not have branded suspension columns, you can turn on the headphones. There is no sound input as such as such - the monitor decodes the sound coming through the HDMI interface.



The 27-inch model adds audio outputs: instead of one stereo, three - front speakers, rear speakers, central column and subwoofer appeared. There is still no analog audio input, so sound on the monitor will have to transmit on an HDMI or DisplayPort interface. Multi-channel sound monitor can only be taken in PCM format, no Dolby format decoder in it.



Finally, the older model, U3011H, adds to the arsenal of the inputs of the second HDMI - and this is perhaps the record among the monitors you have seen! True, a composite video input disappeared, but it is hardly with its low quality signal to anyone will be needed on a 30-inch monitor.

Well, if the differences between the sensory and mechanical buttons were mainly decorative, then a clear positioning is traced in the input set: the more expensive the monitor, the more he can. The number of video inputs only changes from three pieces on younger models up to seven (SIC!) On the eldest. In this case, all models have a DisplayPort input, which only recently began to appear on video cards.

Screen menu and features

The menu of all Dell monitors under consideration today are also performed by one template, but with differences in functionality - that is, in the richness of settings. Therefore, at first I get a little attention to his appearance, and then I will say separately than different monitors differ from each other.


When you press any of the control buttons (four of them on younger models and five on the elders), the shortcut menu opens from which you can switch the input, change the brightness and contrast, select one of the preset modes, and also get into the full settings menu. Points "MENU" and "EXIT" always remain unchanged, but the functions of the upper two (in U2211H and U2311H) or three (in other models) buttons can be at their discretion.



Preset modes - from 6 to 10 pieces, depending on the monitor model; In reality, however, it is unlikely that someone will use more than two three. Dell decided to move into a single list of modes at once all - and typical "image enhancers" ("Multimedia" and "Game"), and the choice of color temperature, and even switching color coverage - for senior models, as in U2211H and U2311H, coverage is possible Only one, SRGB.

Unfortunately, profiles, simply changing the brightness and contrast of the monitor, no - therefore, if "image enhancers" you do not need, their practical usefulness is small.


The brightness adjustment mini menu is quite usually two sliders with a range of 0-100.



The full menu consists of eight bookmarks, in each of which many settings - everything does not have a special meaning in detail, since in most cases their purpose is obvious.



As noted above, two or three buttons, referred to in the menu as "shortcut keys", can be reassigned by the user - on each of them you can "hang" one of the following functions: Select the pre-installed profile, adjustment of brightness and contrast, auto-tuning under analog signal , Enabling the "picture-in-picture" mode (on those monitors where it is) or switching the inputs.

Compare the settings and functionality of different models are convenient, with information about their menu in the table:



"Shortcut Keys" - the number of monitor buttons reassigned by the user.
"INPUT COLOR FORMAT" is a color coding format (why on monitors without a component video input, YPBPR is needed, not very clear).
"Gamma" is an affordable gamma adjustment, for all models it is a choice between 2.2 (PC) and 1.8 (Mac).
"PRESET MODES" - the number of pre-installed settings profiles. Senior models are highlighted in both due to the appearance of switching between the SRGB and AdoberGB color spaces, and due to the additional "enhancer of the picture".
"SRGB" and "AdoberGB" - the oversight of switching color space. Two younger models have standard color coverage and therefore adobergb does not support.
"WIDE MODE" - interpolation modes of the picture in a non-resolution monitor.
"Sharpness" - adjustment of the clarity of the picture.
"Noise reduction" is a mode called, according to Dell, improve the clarity of the edge of dynamic images. Available only in "Game" profiles, "Multimedia" and "Movie".
"Dynamic Contrast" - inclusion of the dynamic contrast mode. Available only in "Game" and "Movie" profiles.
"Line Out Source" - Sound sources for linear output. There are no analog audio inputs.
"Audio Configurations" - possible configurations of an external speaker system connected to the monitor. Recall that the multichannel sound on the monitor can only be transmitted by digital interface (HDMI or DisplayPort) and only in PCM format.
"Picture-BY-PICTURE" is a mode in which two pictures from different inputs are located on the screen next.
"Picture-in-Picture" is a mode in which two pictures from different inputs are located one on top of another.

The table is obvious that the senior models are different and the functionality of the firmware: the larger (and more expensive) monitor, especially since he has the realization of which requires only firmware modifications - more interpolation modes, more preset settings ...

Only U2410 is allocated, which is the only entire line of the Picture-in-Picture mode - display the picture from one input over the picture from the other. The older two models can also work with two inputs, but they are located next to each other in them.

For each of the models that support PIP or PBP modes, different combinations of simultaneously active inputs are possible: D-sub, DisplayPort, component or composite can be combined with any other input, but only one can work out of the available DVI and HDMI dialing. entrance. In other words, if DVI is assigned as the main source of the signal, then you cannot select HDMI as a second source for PBP mode.

In general, the monitor menu can be estimated as convenient in circulation and rather functional for any everyday tasks. The largest minus, perhaps, hides in the sets of preset settings: first, it is not possible to quickly change the brightness of the monitor without affecting its color rendition ("improvements", which are traditionally accompanied by pre-installed modes, do not like many), secondly, some possibilities - For example, dynamic contrast - toughly tied to specific profiles, so if you like the dynamic contrast, but I do not like what makes the "Movie" mode with the color rendition, you will have to either give up the first one or come to compete with the second.

Yet, probably, I will never get tired of repeating: the most convenient implementation of rapidly switched profiles was already invented by Samsung. In its monitors, unfortunately, not all - there are several profiles that differ from each other. only Brightness and contrast and not affecting the color reproduction that is not described by these two parameters, as well as a separate profile, including dynamic contrast. All color settings, including all sorts of "intelligent color reproduction improvers", in the Samsung monitors are made to another place of the menu. Dear monitor manufacturers, please follow this example.

E-IPS Matrix: Pros and Cons

All monitors under consideration today combines one feature - and it should be noted, very important: they are built on matrices such as E-IPS, which have been developing a long time, well-proven itself, but, unfortunately, too expensive S-IPS technology. It is with the main disadvantage of S-IPS - the price - and the new development of the LG company must be fighting, if possible, keeping the advantages of S-IPS.

Pluses S-IPS:

excellent viewing angles both horizontally and vertically: the colors on the screen look almost the same when looking straight and at an angle;
good time Response: S-IPS paired with not too aggressive response compensation scheme shows the result of 6-10 ms (GTG), which makes monitors on such matrices fast for games.

Cons S-IPS:


when viewed at an angle, the black color acquires a characteristic purple shade;
"Crystal Effect": Anti-reflective coating of old models of matrices gave an unpleasant effect, the picture seems to be slightly overflowing; In the new models, they are quite effectively fighting;
low efficiency: small transparency of the matrix requires the use of more powerful backlight;
high price.

Although it would seem that the list of drawbacks is longer than the list of advantages, thanks to only one of them, the S-IPS matrices turned out to be often indispensable: only they guaranteed the minimum distortion of color reproduction when looking at the monitor at an angle, while competing technologies - PVA, S-PVA , C-PVA, TN - the tonality of the picture quite noticeably depended on how to look at it, for example, a blue color could get a clear greenish tint at a side. TN has added more to the minuses. vertical angles The review, PVA - the disappearance of the most dark parts of the image (this effect could be adjusted by the monitor settings, but then a high contrast, one of the serious advantages of PVA) and a great response time ... In general, the S-IPS matrix became actually the only choice for People interested in accurate color reproduction.

Unfortunately, their high cost inevitably trembled the corresponding monitors into a narrow segment of the market of professional and semi-professional, but in any case the extremely lowest models.

To solve this problem is called a new generation of IPS matrices - E-IPS, characterized by another pixel structure and greater transparency that allows you to reduce the required brightness of the backlight, and therefore cost and energy consumption.

Pluses E-IPS:

comparable with S-IPS viewing angles as horizontally and vertically: the colors on the screen look almost the same when looking straight and at an angle;
good speed: When using the response time compensation scheme, monitors on E-IPS are suitable for both games;
almost the complete absence of the "crystalline effect";
low value allowing to withdraw monitors to E-IPS to the average price segment.

Cons E-IPS:

average contrast indicators - at the level of TN matrices, but worse than PVA;
when viewed at an angle, the contrast is very falling - the black color is noticeably light.

Obviously, such a relationship of advantages and minuses unambiguously displays models on E-IPS matrices to a category of interesting users - good characteristics, reasonable price and, the main thing, balance, and the other makes such monitors attractive for people who want to acquire anything better models on TN, but who do not have any particularly critical requirements or preferences, and therefore not ready for satisfaction Requirements to pay a prominent amount. If the monitors on S-IPS basically got into the category "I will give a lot for color reproduction," the E-IPS is rather "a little more money for the best picture": Agree, a much wider market segment.

By and large, E-IPS has only one noticeable inalienable - that is, present on all such matrices - disadvantage: flavoring black when viewed at an angle, effect, in English forums that received its own name "Glow", although in general it is Typical for all LCD matrices dropping contrast, just in this case manifests first in black color.

Regardless of the type of matrix, when viewed at an angle at the LCD monitor, contrast falls: white becomes darker, and black - lighter. However, if both effects are usually about equal in strength, then in the case of E-IPS, the flames of black is noticeably released - even with relatively small angles of view, it turns into dark gray.

Nevertheless, in practice, this effect does not affect the picture: it is manifested only on black color, while the other shades on E-IPS even when viewed at an angle are not simply without special problems, but significantly better than TN or PVA matrices, without noticeable tonal distortion. And of course, even taking into account the airlessness of the black E-IPS in the corners of the review turn out to be two heads ahead of Tn, in which the tonality shift is supplemented with the side of the side more than a noticeable darkening or with the breath of the whole picture when viewed or from above.

To worry about black color on E-IPS stands in one case - if you buy a large monitor first for watching movies in the evening dirty: This is the only scenario of use, in which the depth and uniformity of the black color will be important - in films with an abundance of dark scenes " Threading "black can be noticeable until what if you are sitting close to the monitor, then the corners of the screen will look a little lighter middle. In other cases, it does not deliver any serious trouble.

So, theoretically E-IPS is an excellent choice for people who want to acquire a monitor with better than those common TN matrices, color reproduction, but not ready to pay for this a lot of money nor sacrifice other parameters.

Which is from the point of view of technical parameters, specific monitors on E-IPS matrices produced by Dell, we will now find out.

Dell U2211H: Test Results

Opens our test model with 21.5 "- and, as usual, in its name, the manufacturer rounded this number to 22 (almost all the monitors sold now with screens 21.5", regardless of the type of matrix and the manufacturer, in the title have numbers "22"). The monitor screen resolution is 1920x1080, or how fashionably talk now, FullHD.

By default, the brightness and contrast of the monitor are set to 75%, the level of white 100 cd / sq. M. It was possible to achieve, lowered brightness to 53% and contrast up to 70%. As usual, I can not remember that the level of 100 cd / sq. M., and how we achieve it is not an unconditional manual for setting up your monitor, but only some of the same monitors for all monitors in which I look sufficient whether the adjustment range is not manifested at low contrast and brightness some artifacts, and so on. It is not necessary to strive to repeat these settings, it is unlikely that they will be optimal for your workplace.

The brightness of the monitor is controlled by the power of the backlight lamps at a frequency of 180 Hz.


The maximum brightness turned out to be relatively low - only about 200 kD / sq. M. In practice, however, it will be able to deliver the problems except when using a monitor for games or watching movies in the light-lit room - and in other cases such a brightness is quite enough. To work, normal brightness is usually from 70 to 120 kD / sq. M, depending on the external lighting.

Contrast is not recorded, but not bad: more than 700: 1, at the level of modern TN matrices.


Gamma curves of blue and green colors take close to the ideal, but the red deviates quite strongly up.


Reducing the brightness and contrast to the color reproduction of the monitor practically does not affect.


In the "GAME" mode, the blue color also raised, but in general the color reproduction changed little.


There are no special changes in Multimedia mode. Changes are not noticeable when you turn on these modes and the eye, moreover, the brightness and contrast settings are common with the usual Standard mode - respectively, it turns out that if the "Game" mode has a practical value if only because only in it U2211H can include dynamic contrast, the meaning of the existence of "Multimedia" is not too clear. The user manual does not shed light on this riddle, rubbing the common phrase that each of the modes is loaded into the monitor optimal color rendering settings.



The color coverage of the monitor is quite well coincided with SRGB, although there is no accurate match.





The white point in all available modes is shifted towards the green color - enough so that the greenish shade of the image is good noticeable. You can correct it by switting the monitor to Custom (RGB) mode and manually adjusting the color.

Otherwise, there are no problems with the color reproduction at U2211H: all shades are reproduced as expected, transverse strips on gradients are barely noticeable, there is no serious scatter in the tonality between different levels of gray.



On the black, the average non-uniformity of the illumination is 5.4%, the maximum deviation is 18.3%; On white - the same 5.4% on average and 14.7% at the maximum. The result is not the perfect, but acceptable; In the picture, built according to the measurement results, it can be seen that the monitor is a dark "ears" on the sides of the screen, a small narrow illumination from above and a broader bottom, but there are no critical defects.


The monitor response time depends little on which halftons it switches - we can say that it is evenly fast. The arithmetic average was 8.2 ms (GTG), which means that the U2211H is certainly suitable for dynamic games - it is unhappy with them. Except for very avid players will remain unfortunate, only 2-millisecond TN matrices can save.


The operation of the response compensation scheme does not work out without artifacts, their average value It is 7.7%, and maximum - in the region of 35%. With such numbers, the artifacts are practically not visible, it is possible to detect them only if they specifically look at - and know where to look. For comparison, typical gaming monitors on TN matrices, the blunders of the response compensation scheme can reach 60-70%, and the arithmetic average in modes approaching the TN-matrix to the cherished 2 ms (GTG) can reach 20% and even higher.

Thus, in general, in the face of Dell U2211h, we see a practical and universal monitor, which is well suited for both the house and for the office, and designed for people who are not satisfied with the quality of the image - first of all, viewing angles - on TN matrices, but and not ready to pay for extremely nothesive professional modelswhose capabilities for them are mostly not needed. Cost U2211h - on this moment About 9,500 rubles - significantly higher than that of the monitors on TN, but do not forget not only about excellent viewing angles of the E-IPS matrix, but also a functional design, the input of DisplayPort, 4-port USB-hub ...

By the same parameters and quality settings, the U2211H is a typical model of the middle class - there is a number of minor shortcomings, but they are either insignificant or corrected by a simple manual adjustment of the monitor.

Dell U2311H: Test Results

The following model is a close relative of the U2211H, the exception that the screen added to a diagonal and a half inches. For all other passport parameters, these monitors coincide almost completely (in 2311h per 50 cd / sq. M. higher brightness, and only) - Well, whether they differ in reality, we will now find out.

If we talk about the size of the screen, then for films and games definitely the more - the better, but it depends on the size of the pixel. U2311H has the same resolution of 1920x1080 as the U2211H, which means that the pixels on it are a bit bigger - and it is more likely in plus, since the picture on a 21.5-inch model some people may seem too small. 23 "In this respect, more versatile, and therefore, if you do not know which of the two monitors it is to choose, in this case it is worth bothering to the side of larger.

By default, brightness and contrast are 75% installed; The level of white 100 kD / sq. M. It was possible to achieve, reducing the brightness of up to 50%, and the contrast - up to 56%. The brightness is regulated by the modulation of the backlight lamp at a frequency of 180 Hz.


The maximum brightness turned out to be higher than at U2211H, almost one and a half times - it came close to 300 cd / sq. M, which allows the use of a monitor for any tasks with almost any external lighting. In this case, the adjustment range is enough to reduce brightness to a level suitable for work at night with minimal external lighting.

Contrast decreased slightly, but still remained at an acceptable level - about 600: 1 when default settings. It is about the same range - 600-700: 1 - the contrast is in typical modern monitors on TN matrices, while monitors on PVA are easily ahead and TN, and E-IPS is one and a half or two times, demonstrating a great black color ( At the same time, however, with a number of innate shortcomings).


When settings by default, gamma curves in U2311H and U2211H are practically no different: acceptable, but no more.


Does not change them and reduce brightness and contrast in the monitor settings.


It is not surprising, but in the "Game" mode, the gamma curves became more careful, gathering into a dense group; At the same time, switching to "Game" has opened access to the setting of dynamic contrast. Generally speaking, the case when in the game mode at the monitor color rendering turns out to be better than the standard, rather amazing.


In Multimedia, the curves again scattered slightly, moreover, the purpose of this regime remains incomprehensible at all: he simply does not differ from the "Standard", the brightness and contrast settings are completely common, the dynamic contrast in it is blocked ... there is a feeling The Dell engineers introduced it simply as a kind of placebo: Surely, if the manual says that the picture becomes better, then some percentage of users will seem.


The most interesting discovery was waiting for us in manual configuration mode "Custom (RGB)": Gamma curves suddenly leveled out and stood up in their place!



The color coverage of the monitor as a whole coincides with the standard SRGB coverage, leaving him on red and slightly shifting towards the green.





The white point and the U2311H is shifted towards the green color, not so much like the U2211H, but it is enough that the greenstone picture is visible to the unarmed look. Considering the sudden correction of the form of gamma curves in the manual mode "Custom", I decided to develop success and installed the sliders in the values \u200b\u200bR \u003d 100, G \u003d 94, B \u003d 96 - this allowed to get rid of the greenish shade and bring the monitor color reproduction Perfect, but at least to very good. Just in case, I remind you that your instance of the monitor may differ configuration (thanks to the use of another version of the firmware, another version of the LCD matrix, and the like factors), therefore it is not worth blindly entering the numbers obtained above - always focus on what you see your own eyes.



The average non-uniformity of the backlight on a black background was 6.1%, the maximum deviation is 19.7%; On a white background - 5.1% and 14.5%, respectively. In the pictures built according to the measurement results, it can be seen that the monitor has a lower half of the screen, as well as the angles in the top. However, the result is within the normal range, in practice, with normal exploitation, this level of unevenness does not interfere.


The average response time was 8.2 ms (GTG), while the maximum fixed value was barely exceeded 10 ms - thus monitor, although it does not put records, but quick enough for dynamic games. For comparison, NEC MultiSync EA231WMI, built on a similar E-IPS matrix, but without a response compensation scheme turned out to be more than twice more than twice.


The average of the embossing of the response compensation scheme was 8.6% - the corresponding artifacts (white drivers on a moving image) in most cases are invisible, although if you specifically look closely, you can see them.

If we compare the U2311H and U2211, then the principal differences, in addition to the size of the screen, there are really no - both monitors demonstrate quite good (although not outstanding) parameters and quality settings, and the disadvantages have alone and the same - insignificant or fairly easily corrected. The higher brightness of the U2311H in most cases will not have any meaning for the user - except that he likes to play or watch movies in the brightly lit room, not a rearguing curtains. Thus, it is necessary to choose between these two monitors first in the size of the screen - from our point of view, the 23-inch model is more convenient and universal, but you may have a different opinion. Do not forget that U2311H is more expensive than younger brother - it costs about 12 thousand rubles.

Dell U2410: Test Results

If the first two monitors coincided in most characteristics, then the U2410 is a model already a few different class. First, in addition to an additional inch diagonal, this monitor received the 1920x1200 resolution with the aspect ratio of 16:10, while U2211H and U2311H resolution 1920x1080 - and the aspect ratio of the screen, respectively, 16: 9. Secondly, the U2410 acquired analog video inputs (component and composite), the second DVI and HDMI input, as a result of which it can be connected immediately to five computers, as well as a DVD player and something else. At the side of the monitor next to the USB ports, a card for SecureDigital-flash drives was added, and the buttons from the usual mechanical steel were sensory.

However, I wrote all this at the beginning of the article, and now time to talk about the quality of the image. In this matter, the U2410 also has significant differences from younger models: it uses high-speed lighting lamps with an enhanced spectrum, which should give the monitor advanced color coverage. In the monitor menu, you can switch between three color profiles - AdoberGB, SRGB and native (without software correction).

It should be noted that the monitor is not lucky with color profiles: in the first version of its firmware (in the A00 monitor version - the version number is indicated on the monitor label) The SRGB profile was implemented quite badly, with a strong grainy of the image on dark shades, reminiscent of the image, made digital photograph With a "noisy" matrix. Subsequently, the mistake was corrected, the benefit of it turned out to be purely software. Now it is guaranteed to purchase a fixed firmware monitor, if you take the A01 version, although on the forums some users note that in the last A00 batches of the firmware have already been corrected. In addition, there is also the ability to update the firmware at home, however, it will be necessary to connect the U2410 to the USB to the system with another monitor, otherwise you will not be able to control the progress of the update process.

By default, the brightness and contrast of the monitor were installed by 50%; The level of white 100 cd / sq. M. It was possible to achieve, reducing the brightness of up to 30%, and the contrast - up to 34%. The brightness is regulated by modulating the power of the backlight lamp at 180 Hz.


The maximum brightness of the monitor is very high - more than 370 cd / sq. M, but the contrast fluctuates around typical for E-IPS modest level - 600: 1.


When settings by default, U2410, all three gamma curves take place above ideal, which means that the image will be a little flexible, with low contrast.


Reducing the brightness and contrast in the monitor settings no significant effect on gamma curves has no.


A little better situation becomes in "Game" mode: at least these additional modes in practice are still just as useless, as well as in the two monitors discussed above, say thanks to Dell's company at least for the fact that they really improve a little, and do not worsen color reproduction .


The picture in Multimedia mode does not differ from "Game" - it seems that they generally differ only in that in the second one can include dynamic contrast, and in the first one can not. In the light of this meaning of the existence of Multimedia mode, continues to escape from me.


SRGB color coverage mode is unexpectedly correcting and gamma curves, making them almost perfect.


The same can be said about the emulation of adoberg coverage.



And, by the way, about the color coverage: as the measurements show, it is really very different from the coverage of models U2211H and U2311H. The blue color remained in place, the red became noticeably richer, and the green moved to the left in the diagram, also becoming more saturated - however, because of the same shift, the reproduction of the shades of yellow, part of them was beyond the limits of the color coverage triangle.



Switching to AdoberGB emulation mode shifts the red point, and too much - it turns out to be not on the boundary of the standard triangle of AdobergB coverage, but inside it. At the same time, getting the yellow and yellow-green-green monitor required in AdoberGB are still physically unable, they are outside of its color coverage.



In the emulation of the SRGB-coverage, in addition to the red, the point of green color is shifted - and alas, but the final compliance of the SRGB is noticeably worse than that of U2311H or U2211H, for which this coverage was native, due to the highly shifted left of the top point of the triangle and Minds of shades yellow outside the physical ability of the monitor.

Thus, if you need the most accurate compliance with the monitor coverage with the SRGB standard, the Dell U2410 may not be the most a good choice. If you still stop on it, it will be best to get a hardware calibrator and use the profile created by it in image editing programs, leaving the monitor in the "native" color coverage mode - this will give you a more accurate result at least on red than red. When using the embedded emulation built into the monitor.






And again we see the tendency weigly for the monitors Dell towards the green ... At the same time, although the SRGB and AdoberGB modes correct this shift, the picture in them turns out to be excessively cold - away from the 6500 K., however, in other modes, if you distracted Excessive green and look at the flower temperature, it is not warmer - even in "Warm" raises for 7500 K.

Judging by reports from the forums, in some part of the U2410 this flaw is fixed, and the image on them has become less green and cold. The fact that the manufacturer decided to change the settings can only be welcomed, but even more could be welcomed if he did it in the first part of the monitors, without waiting for buyers' complaints.

It should be noted that the U2410 is like other monitors on E-IPS matrices of LG - sometimes another color defect is found: the right side of the screen has a weak pinkish shade, and the left is greenish. This is a defect of the matrix itself, but not the monitor settings, so manifests itself on models of other manufacturers. The defect in a serious form is relatively rare, but due to its visibility and unusualness is widely discussed in various forums. It is easy to avoid it: when buying, do not forget to check the monitor for color uniformity throughout the screen, withdrawing a simple white or gray field. At the test instance - as well as on other monitors presented in this article - there was no such problem.



But what was, this is the uneven illumination from the left edge of the screen to the right - it is clearly visible on the right picture (I remind you that it is not a photo, and the picture created by the results of the monitor brightness measurements at different points of the screen). Non-uniformity on the black average amounted to 6.3%, a maximum of 13.7%, on white it turned out to be 5.6% and 20.6%, respectively. Unfortunately, such a splice of brightness on the white field from the left to the right edge of the screen was quite noticeable for the eye - although it cannot be said that he greatly interfered with the work.


Dell U2410 is quite fast: the average response time is 6.6 ms (GTG), on a quarter faster than U2211H and U2311H.


At the same time, the average promises of the response compensation scheme grows up to 9.7% - however, and at this level, in practice, the artifacts are sufficiently minorly accommodated.

In general, the U2410 produces an ambiguous impression: according to its capabilities, the monitor could claim professional work with color, but in practice, the quality of its configuration does not rise above the average level. The cost of U2410 is significantly higher than the U2311H - more than 20 thousand rubles will have to give for it.

Is there a U2410 of such money or not? You can answer this question only you yourself - if you need a richest set of inputs and a resolution of 1920x1200, then definitely stands. However, there are no other fundamental advantages of U2410, so if you are satisfied with 23 "screen with 16: 9 proportions and you do not need to connect more computers' pairs at the same time - you can safely purchase U2311H, of course, losing in the size of the screen and capabilities, but at the same time And standing more than one and a half times less. If you count on the most accurate color reproduction, then in a pair to U2410 does not interfere with the hardware calibrator.

Dell U2711: Test Results

The next monitor is an extremely interesting model. I usually dislike 27 "Monitors: due to the resolution of 1920x1080 or 1920x1200, they have a large pixel size and are actually interesting only for games or movies, but they do not exceed any more cheap 24" monitors as working models.

However, U2711 is not from their number. With the amount of 27 inches, this monitor has the same resolution as the 30-inch models - 2560x1440 (aspect ratio of 16: 9). Of course, many will be uncomfortable very small pixel size - 0.233 mm - but, on the other hand, if you take the case requiring output to the screen large number Information, but at the same time you do not have 40-50 thousand rubles for the purchase of 30 "monitor, then u2711 with a price of less than 30 thousand rubles may be a real find. I think readers who are often working in CAD / CAM systems with complex drawings, in the worski programs And another similar software will appreciate my enthusiasm.

By default, the brightness and contrast on the monitor are installed by 50%; The level of white 100 cd / sq. M. It was possible to achieve, reducing the brightness of up to 30%, and the contrast - up to 38%. The brightness is regulated by modulating the power of the backlight lamp at 180 Hz.


The monitor turned out to be very bright: more than 350 cd / sq. M at the maximum. However, brightness without problems is reduced to a comfortable level. The contrast was in the already familiar to other models of the range of 600-700: 1, if not counting the mode with reduced brightness.


Gamma curves are slightly raised relatively ideal, but the deviation is quite small. Without overloading the article by schedules, it simply note that a reduction in the contrast in the monitor settings is, as well as the transition to the "Game" and "Multimedia" profiles, there was no significant effect on the curves form: all the same slight overestimation relative to the calculated curve for gamma 2.2, In practice, expressing in a slightly reduced contrast of the image. Light and dark shades reproduced without problems, there were no transverse bands on gradients noticeably.


As in the case of U2410, switching to SRGB mode leads gamma curves extremely close to the ideal.


The same thing - and in the "AdoberGB" mode.



These two monitors and color coverage profiles are similar: coincidence with SRGB on blue color, superiority on red and top of the triangle left. Because of the last monitor, the monitor is physically not able to reproduce the range of SRGB-colors - some yellow and yellow-green are outside its color coverage.



Profile "AdoberGB" adjusts the position of the red point, and more careful than the U2BERGB, it turns out to be a little further on the border of the adwaregb coverage, while the 24-inch model has got inside the triangle, as a result of which the red playback was worse, What is put in adobergb.



In the SRGB mode, the position of the green point is also adjusted, but since its own monitor coverage initially does not overlap SRGB, then in the emulation mode, it will not be possible to achieve accurate compliance - some shades of yellow will be outside the possibilities.

Thus, although the modes of emulation of various color coverages on the U2711 are set up more accurate than on the U2410, hopefully the ability of this monitor to fully display 100% SRGB coverage or adobergb: This requires that the native monitor coverage completely overlaps the emulated coverages that in this There is no case. Therefore, for whom the most accurate color work is important, even in SRGB or AdoberGB emulation mode, I recommend creating a profile of this monitor using a hardware calibrator and connect it to the image editing program you are using.

If you simply arrange a good color reproduction, then the "SRGB" and "adobergb" modes do not look quite attractive without additional calibration: neat gamma curves and the lack of serious lights in the emulation of color coverage make the monitor color reproduction more accurate than in Standard mode.






Unfortunately, the image on the monitor in all modes is quite cold: only "Warm" dropped to 6500 K, which allows you to call it if not warm, then at least neutral. For comparison, SRGB demonstrates the color temperature of about 8000 K, although the standard for SRGB is precisely 6500 K.

On the other hand, the U2711 cured the problem with the shift of colors towards the green: if in the Standard mode there is still a small deviation from the neutral gray curve (it is shown in black in the diagrams), then the other modes fall on it very precisely.



Like the U2410, on white color on the monitor, a small illumination brightness gradient is noticeable - right part The screen is a bit darker left, and a light horizontal strip passes in the center. If we talk about numbers, then on black color, the average non-uniformity was 3.3% with a maximum deviation of 10.1% ( excellent result!), But with white everything is expected to be worse: the average brightness variation is 7.4%, the maximum deviation is 23.3%.


The response time is on average equals 5.7 ms (GTG), that is, U2711 is one of the fastest monitors on the IPS matrix.


Unfortunately, for it I had to pay a high level of artifacts: on average 15.7%, at a maximum up to 45%. It is not as bad as the game TN monitors, on which there are 70% on individual transitions, they would often achieve 70%, but I would still like to see a smaller level of artifacts for a higher response time - an increase in the last to 7-8 ms (GTG) no real influence The subjective sensations of the user from the speed of the monitor will not have, but artifacts in the form of light shadows in moving objects may be an unpleasant surprise.

As a result, we can say that U2711 is simultaneously allocated from the total mass, and no. On the one hand, relatively inexpensive 27 "Monitor with very high resolution (2560x1440 pixels) is clearly worth a mansion: quite recently there were no such models on sale at all - and they are very interesting for people working with large amounts of graphic information, but not ready to spread 10-20 thousand rubles more for 30 "models. With The other side, the quality and accuracy of the U2711 setting is not allocated among other Dell monitors considered by us on E-IPS matrices: it has a row of flaws that cannot be called critical, but it can be unpleasant. Especially depressing bad uniform illumination on white color and enough high level The artifacts of the response compensation scheme - neither one nor other drawbacks are not corrected by the monitor settings.

Dell U3011: Test Results

The last monitor in today's article is a 30-inch U3011 with a resolution of 2560x1600 (aspect ratio of 16:10). It is noteworthy not only by screen sizes, but also magnificent equipment: just a couple of years ago, when monitoring processors were too weak for full processing of video stream with such a resolution, 30 "monitors were equipped with the only DVI input, could only work in two permissions (2560x1600 and 1280x800) And there were exactly one setting - brightness. U3011, on the contrary, has the richest set of inputs and settings among all Dell monitors discussed today, and not only does not give way to other models with smaller permissions in this matter, but also exceeds the absolute majority of them.

By default, the brightness and contrast of the monitor are installed by 50%, the level of white 100 cd / sq. M. It was possible to achieve a brightness of 35% and the contrast of 36%. The brightness is controlled by the power of the backlight lamps at a frequency of 180 Hz.


It seems that Dell was set to make larger monitors and brighter - U3011 at a maximum approaching 400 cd / sq. M. True, why should he like this brightness - it is not very clear: it is unlikely that someone will buy a 30 "monitor for films (FullHD TV will cost much cheaper, and the resolution of 2560x1600 is not very necessary in films), for the work of the same brightness above 200 cd / sq. M. Simplicity is not required in any conditions. However, fortunately, the brightness of the monitor can be easily reduced to comfortable values \u200b\u200bat which it will not illuminate the room and make his eyes.

The contrast of the monitor, alas, though quite a bit, but did not reach up to 600: 1.


When settings by default, gamma curves look perfectly, most of the graphic, almost merging with the calculated curve for Gamma 2.2. Reducing the contrast in the monitor settings on their form practically does not affect. The monitor reproduces the entire range of shades, from the most dark to the brightest, transverse strips at the gradients is not visible.


In AdoberGB mode, the shape of the curves is mainly maintained, only a little more becomes disordered in the field of dark tones - but also the eye it is not noticeable there, and the accuracy of the calibrator in this part is small.


Picture in SRGB mode matches almost pixel-in-pixel.



The native color coverage of the U3011 is the same as in the previous two models: it is ahead of SRGB on red and green colors, but due to the deviation of the vertex of the triangle, it is not capable of overlapping the entire SRGB coverage entirely, lagging behind it on the shades of yellow. Therefore, to expect from the U3011 of the accurate emulation of the standard SRGB coverage, it is not worth such an option in the menu and there is on a par with an adobergb emulation.



In the "AdoberGB" mode, the monitor will noticeably adjust the position of the red point and slightly green, with the result that its color coverage does not go beyond the standard coverage of AdoberGB. True, at the same time he Adobergb and does not cover - yellow and even partly red colors in the AdoberGB standard is cleaner and rich than it is able to show U3011.



The same story and with the "SRGB" mode: the position of the red and green points is adjusted so that the final coverage of the monitor does not go beyond the SRGB limits, however, and cover the entire range of colors SRGB monitor is not physically capable - no correction program will allow to circumvent restrictions laid down in Luminofor lamp lighting.






As with other Dell monitors, discussed today, the color temperature in the U3011 is noticeably overpriced, the picture is too cold, even the "SRGB" and "adobergb" modes instead of the 6500 K area turn out to be around 8000 K. On the other hand, the monitor in no one There are no modes for some noticeable deviation into green or pink tones, and temperatures different levels Gray in most cases lie close to each other.

By the way, a leaf with the results of measurements of the color temperatures of this instance, also clearly demonstrating the good neutrality of gray, is inserted into the box with senior models of Dell monitors. True, it is not clear why with so thorough individual approach Dell engineers did not provide for calibration not only on color temperature spreading, but also in its absolute value.



Although non-uniformity on white color is quite well noticeable, the monitor does not have such a pronounced brightness gradient from left to right, as in U2410 and U2711 - and therefore, the eye with the usual work this defect is noticeable. If we talk about numbers, then on black color, the average non-uniformity of the backlight was equal to 4.6% at a maximum of 12.3%, on white - 6.9% and 20.4%, respectively.


Fortunately, in the case of the U3011, the developers did not chase the records - and the average response time is 8.4 ms (GTG), which is quite enough for games, and for films.


At the same time, the flaws of the compensation scheme of the response are actually absent - they are in just a few transitions, therefore the average value of the emboss is only 0.6%. IN real life, without special measuring equipment, the artifacts corresponding to these misses will be simply impossible.

As a result, from the older three monitors of Dell - U2410, U2711, U3011 - it was the 30-inch model that was the most neat in terms of settings: the correct color rendition (except for the tendency to cold colors), the lack of asymmetric illumination of the screen, good response time with actually missing Artifacts ... However, there is nothing surprising in this: the retail value of the U3011 in Moscow exceeds 50 thousand rubles (we recall that the U2711 can be found in less than 30 thousand).

However, if you are ready to spend on the monitor such an amount, then the U3011 will be an excellent choice: a neat appearance, a magnificent functionality, wealth of inlets and opportunities for all occasions, as well as neat adjustment will not leave you disappointed.

Conclusion

Summing up, it can be said that the miracle did not happen: Dell monitors, positioned in the average price segment - is out of this except that the inhabited U3011 - demonstrated the possibilities and quality characteristic of the average price segment.

Two younger models, 21.5 "U2211H and 23" U2311H, produced in general a very favorable impression: good monitors with a convenient menu, a complete set of adjustments and a neat appearance perfectly suitable for recreation, and for work. Their higher price compared to models on TN matrices is quite justified functional design And using the E-IPS matrix, released by its viewing angles. These two models are not an interior decoration, but excellent "working horses", which you will not be disappointed now, no one after a year or through three.

If we talk about the choice between u2211h and u2311h, it comes down to the choice between the price and extra one and a half inches of the screen - there are no other significant differences between these two models. I consider U2311H more convenient and versatile, but your opinion may differ from mine.

The 24-inch U2410, on the contrary, has become a small disappointment: from the model worth more than one and a half times higher than the U2311H, you expect not only an extra inches of the screen and a pair-triple video inputs, but first of all higher quality settings. Even if you do not remember that AU00 revisions are not lucky with the firmware, then in the version A01, the monitor setting, albeit from very clear errors, still leaves much to be desired: not very neat gamma curves, strong non-uniformity lighting, shift of the tonality to the side Green color, not very accurate emulation of the standard covers of SRGB and AdoberGB ... Perhaps, if you do not have acute needs in the resolution of 1920x1200 (instead of 1920x1080), one excess inch of the screen and additional video outputs, I would advise saving money and purchase U2311H, as the demonstrating not worse Quality with a much lower cost.

Despite the fact that the 27-inch U2711 was also unable to distinguish between the high accuracy of the settings, this monitor is still special attention: at a reasonable value (less than 30 thousand rubles), it has almost the same resolution as much more expensive 30 "models - 2560x1440. Such a model can be a real salvation for people working in CAD / CAM programs, layers and the like-like software for which there is no too large screen.

Finally, 30 "Dell U3011 is just a good monitor, deprived of a significant part of the shortcomings of his younger brothers, neatly configured and at the same time possessing magnificent functionality. If you are ready to spend more than 50 thousand rubles - this model will become an excellent choice.

If we talk about the E-IPS matrix in general, then this technology has made a pleasant impression with its capabilities - despite such disadvantages, as a relatively low contrast and flavoring black when viewed at an angle, it significantly exceeds TN in the corners of the review and the quality of color reproduction, can provide Sufficient response time for games and films, and most importantly, lowers monitors on IPS matrices from heaven to the ground, in the middle price category.

However, the manufacturer of the LCD panels would be worth a little tighten the quality: in two of the five monitors I ran into such an unpleasant phenomenon, as a noticeable non-uniformity of the backlight from the left to the right edge of the screen, the forums regularly distribute complaints about a tonal shift in different parts of the screen aside Green I. pink flowers... These are defects of LCD panels, not monitors, and they are found not only from Dell, but also from other manufacturers using E-IPS matrices. Of course, a careful inspection of the monitor when buying will allow to avoid such problems, however, first, it is not always possible, secondly, in any case, I would like to at least be able to do without it without risking at the same time to get a defective monitor.

However, I can not fail to repeat that the overall impression of E-IPS matrices, no matter what, positive. Not only were they able to make quite affordable monitors on matrices that differ highly different from TN, so also stirred by the industry - and in the next article we will look at the answer to E-IPS the second largest manufacturer of LCD panels, Samsung, - monitor Syncmaster SA850 on the PLS matrix. Appointment of service. The matrix calculator is designed to solve systems linear equations The matrix method (see an example of solving such tasks).

Instruction. For online solutions It is necessary to choose the form of the equation and set the dimension of the corresponding matrices.

View of the equation: A · x \u003d b X · a \u003d b A · x · b \u003d c
Matrix dimension A.
Matrix dimension B. 1 2 3 4 5 6 7 8 9 10 x 1 2 3 4 5 6 7 8 9 10

Matrix dimension C. 1 2 3 4 5 6 7 8 9 10 x 1 2 3 4 5 6 7 8 9 10

where A, B, C - the specified matrices, x - the desired matrix. Matrix equations of the form (1), (2) and (3) are solved through the inverse matrix A -1. If the expression A · X is B \u003d C is specified, then it is necessary to first fold the matrices C + B, and find a solution for expressing a · x \u003d d, where d \u003d c + b (). If the expression A * x \u003d b 2 is specified, then the pre-matrix B must be erected into a square. It is also recommended to familiarize yourself with the main actions above the matrices.

Example number 1. The task. Find a solution to the matrix equation
Decision. Denote:
Then the matrix equation will be recorded in the form: a · x · b \u003d C.
The determinant of the matrix A is DETA \u003d -1
Since a non-degenerate matrix, then there is a reverse matrix A -1. Multiply to the left of both parts of the equation on A -1: Multiply both parts of this equality on the left to A -1 and on the right to b -1: a -1 · a · x · b · b -1 \u003d a -1 · c · b -1 . Since a · a -1 \u003d b · b -1 \u003d e and e · x \u003d x · e \u003d x, then x \u003d a -1 · c · b -1

Reverse matrix A -1:
Find a reverse matrix B -1.
Transposed Matrix B T:
Reverse matrix B -1:
The X matrix is \u200b\u200blooking by formula: x \u003d a -1 · c · b -1

Answer:

Example number 2. The task. Solve matrix equation
Decision. Denote:
Then the matrix equation is recorded in the form: A · X \u003d B.
The determinant of the matrix A is equal to Deta \u003d 0
Since A degenerate matrix (the determinant is 0), therefore the solution equation has no solution.

Example number 3. The task. Find a solution to the matrix equation
Decision. Denote:
Then the matrix equation is recorded as: x · a \u003d B.
The determinant of the matrix A is equal to Deta \u003d -60
Since a non-degenerate matrix, then there is a reverse matrix A -1. Multiply to the right of both parts of the equation on a -1: x · a · a -1 \u003d b · a -1, from where we find that x \u003d b · a -1
Find a reverse matrix A -1.
Transposed Matrix A T:
Reverse matrix A -1:
X Matrix We are looking for according to the formula: x \u003d b · a -1


Answer:\u003e

Inverse matrix - Such the matrix A. −1 , when multiplying, the original matrix A. Gives as a result single matrix E.:

Square matrix reversible then and only when it is nondegenerate, that is her determinant Not equal to zero. For non-commercial matrices and degenerate matricesinverse matrices do not exist. However, it is possible to summarize this concept and introduce pseudo-male matricessimilar to back to many properties.

The solution of matrix equations

Matrix equations may look:

Ah \u003d in, ha \u003d in, agv \u003d s,

where A, B, C is the specified matrices, which is the desired matrix.

Matrix equations are solved by multiplying the equation for inverse matrices.

For example, to find a matrix from the equation, you must multiply this equation to the left.

Consequently, to find the solution of the equation, you need to find a reverse matrix and multiply it to the matrix, standing in the right part of the equation.

Similarly, other equations are solved.

Example 2.

Solve the equation ah \u003d in if

Decision: Since the reverse matrix is \u200b\u200bequal (see Example 1)

Linear spaces

Determination of linear space

Let be V. - Non-empty set (its elements will be called vectors and denoted ...) in which the rules are installed:

1) any two elements correspond to the third element called the sum of the elements (internal operation);

2) Each and each corresponds to a certain element (external operation).

Lots of V. It is called valid linear (vector) space if axioms are performed:

I.

III. (zero element, such that ).

IV. (element opposite to element), such that

V.

VIII. Similarly, a complex linear space is determined (instead of R. Considering C.).

Subspace linear space

Many are called the subspace of linear space V., if a:

1)

Linear space system system L. Forms basis in L. If this system of vectors are ordered, linearly independent and any vector of L. linearly expressed through system vectors.

In other words, a linearly independent ordered system of vectors e. 1 , ..., e. n. forms Basis B. L. If any vector x. of L. can be presented in the form of

x. \u003d C 1 · e. 1 + C 2 · E. 2 + ... + with n. · e. n. .

You can define the basis otherwise.

Any ordered linearly independent system e. 1 , ..., e. n. vectors n-dimensional linear space L. n. Forms the basis of this space.

Insofar as n., Space dimension L. n. - Maximum number of linearly independent space vectors, then system of vectors x.,e. 1 , ..., e. n. linearly dependent and, therefore, vector x. linearly expressed through vectors e. 1 , ..., e. n. :

x. = x. one · e. 1 + x. 2 · E. 2 + ...+ x. n. · e. n. .

Such a decomposition of the basis of the basis only.

Theorem 1. (On the number of vectors in linearly independent and generating vectors.) The number of vectors in any linearly independent system system does not exceed the number of vectors in any vector generating system system vector space.

Evidence. Let an arbitrary linearly independent system of vectors, an arbitrary generating system. Suppose that.

Because The generating system, then it represents any space vector, including the vector. Attach it to this system. We get a linearly dependent and generating system of vectors: . Then there is a vector of this system, which is linearly expressed through the previous vectors of this system and it, due to lemma, can be removed from the system, and the remaining system of vectors will still generate.

Purge the remaining system of vectors: . Because This system is generating, it represents the vector and, connecting it to this system, again we get a linearly dependent and generating system :.

Next, everything is repeated. There is a vector in this system, which is linearly expressed through the previous ones, and this can not be vector, because The source system is linearly independent and the vector is not expressed linearly through the vector. So it can only be one of the vectors. Removing it from the system, we get after reprimion, the system that will generate the system. Continuing this process, after steps we obtain the generating system of vectors:, where, because In our assumption. It means that this system, as generating, represents both the vector, which contradicts the condition of linear independence of the system.

Theorem 1 is proved.

Theorem 2. (On the number of vectors in the base.) In any base vector space contains the same number of vectors.

Evidence. Let both - two arbitrary vector space bases. Any basis is a linearly independent and generating system of vectors.

Because The first system is linearly independent, and the second is generating, then, according to Theorem 1 ,.

Similarly, the second system is linearly independent, and the first is generating, then. It follows that, bt.d.

Theorem 2 is proved.

This theorem Allows you to enter the following definition.

Definition. The dimension of the vector space V above the field K is the number of vectors in its base.

Designation: or.

Coordinates of the vector - the coefficients of the only possible linear combination basis vectors In the selected coordinate systemequal to this vector.

So, services by solving matrices online:

Working with matrix allows you to perform elementary conversions of matrices.
If you have a task to perform a more complex transformation, then this service should be used as a designer.

Example. Dana matrix A. and B., need to find C. = A. -1 * B. + B. T,

  1. You should first find reverse matrixA1 = A. -1, taking advantage of the service to find the inverse matrix;
  2. Next, after found a matrix A1 performed matrix multiplicationA2. = A1 * B.using the multiplication service of matrices;
  3. Performed transposing the matrixA3. = B. T (service for finding a transposed matrix);
  4. And last - we find the amount of matrices FROM = A2. + A3. (service for calculating the amount of matrices) - and get the answer with the most detailed decision!;

Work of matrices

This is online service in two steps:

  • Enter the first factory matrix A.
  • Enter the second factor of the matrix or vector column B.

Multiplication of the matrix on the vector

Multiplication of the matrix on the vector can be found using the service Matrix multiplication
(The first factory will be this matrix, the second factory will be a column consisting of elements of this vector)

This is online service in two steps:

  • Enter the matrix A.for which you need to find a reverse matrix
  • Get an answer with a detailed solution to find the return matrix

The determinant of the matrix

This is online service in one step:

  • Enter the matrix A.for which you need to find the determinant of the matrix

Transposing the matrix

Here you can track the matrix transposition algorithm and learn to solve such tasks yourself.
This is online service in one step:

  • Enter the matrix A.that you need to transpose

Rank matrix

This is online service in one step:

  • Enter the matrix A.for which you need to find the rank

Own matrix and own matrix vector

This is online service in one step:

  • Enter the matrix A.for which you need to find your own vector and own meanings (Own numbers)

Construction of the matrix to the degree

This is online service in two steps:

  • Enter the matrix A.which will be raised to the degree
  • Enter an integer q. - degree

The matrix is \u200b\u200ba mathematical object recorded in the form of a rectangular table of numbers and allowing algebraic operations (addition, subtraction, multiplication, etc.) between it and other similar objects. The rules for performing operations on matrices are made by such

to make it convenient to record system of linear equations. Usually, the matrix is \u200b\u200bdenoted by the capital letter of the Latin alphabet and isolated round brackets "(...)" (found also

the selection by square brackets "[...]", double straight lines "|| ... ||") and the numbers constituting the matrix (elements of the matrix) are denoted by the same letter as the matrix itself, but small. Each element of the matrix has 2 lower index (A ij) - the first "I" means

the line number in which the element is located, and the second "j" is the column number.

Operations on matrices

Multiplication of the matrix A for the number

B, the elements of which are obtained by multiplying each element of the matrix on this number, that is, each element of the matrix is \u200b\u200bequal to

b ij \u003d λ a ij

Addition of the matrices A.

element matrix C is equal

c ij \u003d A ij + B ij

Subtraction of matrices A.

c ij \u003d a ij- b ij

A + θ \u003d a

Matrix multiplication(Designation: AB, less often with a multiplication sign) - there is an operation for calculating the matrix, the elements of which are equal to the amount of the pieces of elements in the corresponding line of the first factor and the second column.

c ij \u003d Σ A IKB KJ

In the first multiplier, there must be as many columns as rows in the second. If the matrixa has dimension, b -, then the dimension of their workab \u003d C

there is . Multiplication of matrices is not commutative. This can be seen at least from the fact that if the matrices are not square, then you can multiply only one to another, but not vice versa. For

square matrices The result of multiplication depends on the order of the factory.

Only square matrices can be erected.

Single matrix

For square matrices exists single matrixE such that multiplication of any

the matrices do not affect the result, namely

EA \u003d AE \u003d A

In a single matrix, the unit is only

diagonal, the remaining elements are zero

For some square matrices, you can find the so-calledreverse matrix.

Reverse matrix A - 1 is such that if you multiply the matrix on it, then a single matrix will be

AA - 1 \u003d E

Reverse matrix exists not always. Matrices for which the opposite exists is called

non-degenerate, and for which there are no - degenerate. The matrix is \u200b\u200bunreath, if all its lines (columns) are linearly independent as vectors. Maximum number of linear independent lines

(columns) is called the rag of the matrix. The determinant (determinant) of the matrix is \u200b\u200bcalled normalized asymmetric linear functional on the lines of the matrix. The matrix

degenerate then and only if its determinant is zero.

Properties of matrices

1. A + (B + C) \u003d (A + B) + C

2. A + B \u003d B + A

3. A (BC) \u003d (AB) C

4. A (B + C) \u003d AB + AC

5. (B + C) A \u003d BA + CA

9. Symmetric matrixA positively defined (a\u003e 0), if the values \u200b\u200bof all its main corner minorors k\u003e 0

10. Symmetric matrixA negatively defined (a< 0), если матрица (−A )

it is positively defined, that is, if for any K chief minor k-th correction K has a sign (- 1) k

Systems of linear equations

Cynoma from M SN equations unknown

a11 x1 + a12 x2 + ... + a1n xn \u003d b1 a21 x1 + a22 x2 + ... + a2n xn \u003d b2

am x1 + am x2 + ... + am xn \u003d bm

can be represented in matrix form

and then the entire system can be written like this: AX \u003d B

Operations on matrices

Let A ij be the elements of the matrix, AB IJ - matrices.

Multiplication of the matrix A for the numberλ (designation: λa) is to build a matrix

B, the elements of which are obtained by multiplying each element of the matrix on this number, that is, each element of the matrix is \u200b\u200bequal to ij \u003d λa ij

We write the matrix A.

Multiply the first element of the matrix A on 2

Addition of the matrices A.+ B is the operation of finding the matrix, all the elements of which are equal to the steamy sum of all relevant elements of matrices IB, that is, each

element matrix C is equal

c ij \u003d A ij + B ij

A + in writing the matrix A and in

Perform the addition of the first elements of the matrices

Spread the values, first horizontally, and then vertically (on the contrary)

Subtraction of matrices A.- B is determined similarly to addition, this is the operation of finding the matrix, whose elements

c ij \u003d a ij- b ij

Addition and subtraction is allowed only for matrices of the same size.

There is a zero matrix θ such that its addition to another matrix A does not change a, that is

A + θ \u003d a

All zero matrix elements are zero.

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