if-a-is-a-6-times-4-matrix-what-is-the-smallest-possible-dimension-of-nul-a

Question

If $A$ is a $6 	imes 4$ matrix, what is the smallest possible dimension of Nul $A ?$

Bobby Barnes · Accepted Answer

So in order for us to determine what this ball is possible dimension of Knoll a could be, um we can use the following so we can use the fact that we know the ring. So how do you still drink? Rank of a plus? The dimension of the knoll of a should be equal to S O over here. If this is m by in, it should be equal to end. So if we can figure out well, what should our maximum rank of the speech then we can use that to find our Matt or minimum dimension because Max Rank is saying the same thing as minimizing are no a dimension. Let&#x27;s go in and just draw six by four matrix. So 123456 Um, so six rows and then four columns everywhere with the red and green light intersect each other we have on entry. So the maximum rank we could have for this is going to be where we have four pivots here. So we have a pivot in each of the columns, so that implies that Max Ring is going to be or eso if we use that appear. So that&#x27;s gonna be four. Plus the dimension of Noel A is equal to well, in this case in its four. And well, if we subtract that over, that implies our minimum dimension of no a can be zero. So I should probably say men here. Yeah, so that is going to be the smallest possible that we can have.

Subham Jyoti Mishra · Answer

this goes in, we have been given that a six by eight mattress. Now the in order to find Dr Diamonds and off non space. Yeah, we&#x27;ll use around here. Um, so it should be equal in minus, uh, Frank? Well, yeah. This is your man Menace. German Central column space stated now in orderto have this minimum. This will do. This valley is constant is very sorry. Maximum no end is in the form of him cause and so and number of problems. This is eight. So minimum the national. Mm hmm. Is it? 89 s? Make something damaging about the neighborhood. The six? Yeah. Change number off Rose is six. So the next thing that blew off by what columns we do not like civil six. There&#x27;s rank Mexico. There&#x27;ll be six advances.

Subham Jyoti Mishra · Answer

the question. We have been given a methods here, which is by by six metrics. And we have been given that the diamonds, you know, space a quarter. For now, we have been asked to find off the diamonds and off problem space here. This is equal to rank. Okay. By rank Children, we have, uh and is it Jordan diamonds and all? Nothing. Hey, let&#x27;s rank up here. So therefore there any number off problems with his in the form of him? Cross him. So anything over six years 63 44 less the rain off here which employs the bank off here. It&#x27;s important to therefore the dimension off column space K is toe this two dimensional.

Runpeng Li · Answer

in this problem, we have Ah, matrix. That is five by six. Bye bye. 68 drinks. And we know that dimension off the nose. Face up. These matrix is five. Sorry for. And we need to figure out the dimension of the rose face A game. We apply the rank, the room first. So the rank off, eh? ISS and minus invention off. Announce yourself, eh? So that&#x27;s six minus four, which is, too, and that it is equal to do you mention up the rose face of a so

Bobby Barnes · Answer

so they want us to determine the dimension of the row space for these two kinds of maitresse easier. So, um, if you&#x27;re ever unsure, let&#x27;s just go ahead and draw an example. So a four by three matrix is going to have four rows in three columns, and so the way I&#x27;m drawing this, each place where there&#x27;s an intersection, it will have a entry. Now the room space, remember, is supposed to be telling us. Well, like if we were to reduce this, the basis we would kind of get would be, whichever one&#x27;s air non zero on in this case, notice how we could have at a maximum of three Hibbitts here, which would mean then we could clear out this very bottom road to make it all zeros. So the basis would be formed by these top three. If we could do this, which would imply that our Max dimension of Row A is going to be equal to three Now there&#x27;s two ways we could do this next one. I don&#x27;t remember if they say this is the book or not, but notice how this is really just the transpose of it. So we went from four by 3 to 3 by four. So this row space dimension down here should be the same or at least the maximum as what we had over here. And we&#x27;ll see that in a second when we actually draw this out because I don&#x27;t remember if they state that in the book or not, at least quite yet. Um, but we&#x27;ll use that idea here. So let&#x27;s go ahead and transpose it in some sense. So three rows, four columns and then again knows here we could have three pivots, so we&#x27;d have where all these could be a full rank, which then also implies the max dimension of Row A is going to be equal to three. Yeah, so, again, I don&#x27;t quite remember if they said this thing about the transposed in the book yet, but you can see how it still leads us to the same conclusion

Runpeng Li · Answer

All right, so in this problem, you have matrix. That is seven by six. And we know no space estimation. That&#x27;s dimension, huh? Fire. So we need to find out the dimension of column space. Okay, so the dimension of the columns face, remember our note or in problem? Six. That is a dimension of the column space. Hey, it&#x27;s same as the rank obey. And at the same time, the rank by the lengthy REM rank up, eh? Is and minus dimension up the north face. Okay, so in this case, six dimension of space five. So the column space should have mentioned one.

Bobby Barnes · Answer

in this problem we have a matrix that is a six by four. Matrix six x 4. And we want to find out the smallest possible dimension of no space of a. So first thing we need to note here is that the dimension of the road space. Okay. Is smaller or equal to the minimum Of six and 4 by our previous problem. So this is actually for so that excuse me. So that means the rank rank of a Smalling or equal to four. And by the rank theorem given textbook. So we have dimension of the north space. Okay? It&#x27;s equal to and minus rank A. So that is four minus four. So that is three actually. Um Yeah the rank the rank in this case we don&#x27;t just take the four. We have to know why we need to take the four. It&#x27;s because we have to make the non space has a dimension as as less as possible. So we have to make rank as large as possible. So the largest we can take is four. So it&#x27;s a And his counsel and that is four. So minus drank a As large as possible. We can take four for maximum. So that is zero. So that&#x27;s the minimum you mentioned. We can get for the north face of a

Problem

In Exercises 17 and 18 , $A$ is an $m \times n$ m…

Problem 16 Medium Difficulty

If $A$ is a $6 \times 4$ matrix, what is the smallest possible dimension of Nul $A ?$

Answer

$N u l A=0$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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$N u l A=0$

Linear Algebra and Its Applications

Anna Marie V.

Caleb E.

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Vectors Intro

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Caleb E.

Kristen K.

Samuel H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications