in-exercises-17-and-18-a-is-an-m-times-n-matrix-mark-each-statement-true-or-false-justify-each-ans-2

Question

In Exercises 17 and 18 , $A$ is an $m 	imes n$ matrix. Mark each statement True or False. Justify each answer.
a. If $B$ is any echelon form of $A$ , then the pivot columns of $B$ form a basis for the column space of $A$ .
b. Row operations preserve the linear dependence relations among the rows of $A$ .
c. The dimension of the null space of $A$ is the number of columns of $A$ that are not pivot columns.
d. The row space of $A^{T}$ is the same as the column space of $A .$
e. If $A$ and $B$ are row equivalent, then their row spaces are the same.

Runpeng Li · Accepted Answer

problem problem. 18. We start with part. Hey, it says, um if bees any edge inform off A than the people call himself, it be former basis for the column. Space of A. So this Damon actually is false. The reason is that the column space the basis optimum off the com space off A is actually not people Columns A B is the corresponding columns in a or his funding columns in a. So, actually, we&#x27;re taking the columns of A instead of the national home. Okay. Okay. So the next R B, we have Roe operations preserved the win your dependence relations about the rose off, eh? When it isn&#x27;t stroke, all&#x27;s well didn&#x27;t steal false. Why? Well, because we we&#x27;ve performed the, uh We performed the road operations we interchange. You didn&#x27;t inter change in the rose. Then everything will be messed up. Interchanging, rose, mess up things right? Because we are interchangeable. Rosen. We cannot tell which Rosie. It&#x27;s exactly the the linear dependence relations we want we&#x27;re looking for Okay. Hard to see so exas. The dimension of the non space of a is the number of columns of a that are not people. Columns Well, this this statement is true. You guess by our wrecked the room we have mentioned up in all space off a is and minus drank a So if we know the dimension of the no space of a number of the columns off, eh? There are not people columns, but rank of A actually keeps our the number of people columns. So those are not people. Columns is actually the columns that the total comes minus the people call him. So this is a you correctly that not people columns. All right. So hearty, it says means the rose space up. Hey, transposed the same as a common space of a Well, this is also true because that&#x27;s how we perform the transpose operation. Just a bite transpose operation. Okay, Because by the transpose operation, the rose becomes the columns, columns becomes the rose. So the the rose face the game will be the same as a con space. When you do the transpose. Okay, our e So it says if a and B are really equivalent, then they&#x27;re roast bases are the same. But it&#x27;s also true. The reason is that you know the what we have to really prevalent people in the matrices that it&#x27;s actually how we do how we find tree, how we find Rose face so we don&#x27;t

Subham Jyoti Mishra · Answer

in this question. We have a Marcus. This came close and metrics. And we have a set off statements which I read up to Workforce. You will check them one by one. What statement is the rule? Space off is the same as the common space. Off transpose. No, those space off is a total carbon. Yes, off a transport. This is true. Hmm. Programs was off road space off. Is the total number off non zero this panel? They&#x27;re not non zero, though. Viktor&#x27;s. Uh huh. Metrics. But when is transports toe transport that always become columns? So therefore, this is equal and well by work. Government. So the So there is this true second statement? Is he? He&#x27;s in a form of here and his 39 year groups. Because do those people on the beach who a now here, living under be used, uh, take alone normal here. Not really different up here. And and yeah, and that number off non zero of those visible the tree, so it may not be necessary that the Post she rose? Uh huh. Norm zero. Those two roles, maybe, uh, independent. So therefore, we cannot create necessarily say back. Uh, well, studios mama B. C&#x27;s off the circus sports? No. From the whole shipment we have when the dollar statement. We have the diamonds channel. The rose space up here is equal to payments. Um, off column space off here, given he iss not a square metrics. And it is true from the rank here, Um, and also the diamonds and off role space here is equal number. All none. Zero rules in the assist. Ekotto&#x27;s number four columns. So therefore, number all favorite problems. So that days ago, diamonds and off problem space, then in the pork statement we have is some of dimensions of the rules. No space off going toe, remember, Oppose. Mm. Which is diamonds and, uh, no. Is this dimension of Cruz, please? Here has your work diamonds and most is here less diamonds and all column space care. Since dimension of those places, dimensional space here has thank off by rank and which is number off columns is a nautical EMS response with this statement. Mhm miss. On a computer role, prisons can change the train called metrics. You are the true on computer He did can be his grounding. Yeah, So the word is true There in the the rank of metrics can change enroll prisons and computer

Bobby Barnes · Answer

right. So they want us to determine if these statements are going to be true or falls. Um, so this first one, it says essentially, that we have the zero vector and H. So this is a subspace of our in and this is going to be false because it doesn&#x27;t say anything about the other two properties that we need to have, which is it Closed under sums and closed under scaler multiplication. So false. Because we do not No. If actually, let me beat the Patriots. Bigger. Do not know if closed under some and scaler multiplication. Okay, so we have a there now for B. They say we have this set of factors, and the set of all linear combinations is a subspace. And so this one is going to be true, because if we just think about, um, what this is so it would be something like see one V one plus c two, b, two plus dot, dot dot all the way up to C p v p. So we can get the, um zero vector by saying all see, I equal to zero for every eye. So that&#x27;s fine. Um, we can get any scaler out of this by saying, see, hi equals zero except for some value. Um, but I is equal to and where it is, like the vector that we want. So that gives us the scale of property and three, some wolf. I mean, some just kind of comes from the definition. So some is, um, true by definition of linear combination linear combo. So that&#x27;s how we end up getting true for B. See, I believe this one is also true, but let&#x27;s go ahead and kind of draw it out. So if we have over here some Mbai in Matrix, remember what the knoll space really is? It&#x27;s saying, Well, what collections of vectors here output the zero vector. And, um, in order for us to multiply these we needed in by one matrix, which means we would have in entries here and then this is an element of our in. So then the null space has to be a subspace of our because it will have the zero vector on. We&#x27;ll also have where if we add two things by the linearity of matrix, that should be fine. And then if we just multiply this, that should also be fine. Um, well, actually, let&#x27;s go ahead and write all those out. So eso let&#x27;s say just so we can kind of be a little bit mawr exact right? So Ah, we have some matrix A with the vectors. Let&#x27;s just call it X Y elements of the knoll of a All right. So if we want to do the some property So let&#x27;s look at a Times X plus y Well, then this is going to be a X plus a y and by definition of both of these being in the null space, this is the zero vector plus zero vector which is zero vector. And so then that implies X Plus y is also an element of nor a Okay, so we have that one and then first scale er&#x27;s. So this is some. So that&#x27;s good. I guess we should have also did the zero vector. But that one&#x27;s kind of true just by definition. So this is zero element of Norway. Alright, so that&#x27;s good so far and then lastly we need these scaler. So we have a times c of X, remember axes so in the north space. But we can go ahead. And do you see? Actually me, Right This over here. So this is going to be C a X like that so we can apply community since he is just some constant. And then we&#x27;ll that c times zero vector, which is equal to zero vector. So then that implies C X is also in the knoll. So then the constant multiple or the scaler also holds. So, um, I don&#x27;t know if you really need to be that exact, but, I mean, it doesn&#x27;t really hurt now for D the column space of Matrix A is the set of solutions. A X is equal to be all right. And so this is actually going to be, uh, false. And the reason why this is false is because it&#x27;s not necessarily just for the solution of one. It&#x27;s supposed to be the span of all of the columns I&#x27;m sorry of. Ah, yeah, the span of all of the columns. Um, so, yeah, that&#x27;s why this one isn&#x27;t necessarily true. Because it&#x27;s not talking about a solution set and then for E. So this is saying that B is the echelon form of a matrix A in the pivot columns of B form a basis for a And so this one, um, is also false Because it&#x27;s not the pivots of B, but the pivots of call a Arab the pivots of a instead. So this should actually say a here on beacon, actually kind of show that off on the side really fast. Eso Let&#x27;s come down here and do that, right? So let&#x27;s say we have some matrix here. And so this is kind of like you can use, like, a counter example as to why this wouldn&#x27;t be the case. Um, so we have, like, 123 uh, then 567 13, 14, 15. And then, I don&#x27;t know, negative 112 So hopefully if I reproduce this year Ah, this gives me 111 000 And if it doesn&#x27;t just kind of tweak this so it does. But I believe this will give us that. Okay, well, if we look at it, this is a and then this is the echelon form. Be over here. Well, if we were to look at this, though, over here, this is saying if we take each of these are fourth position of our vector is always going to be zero. But over here, notice There are numbers here. So we wouldn&#x27;t even be able to get out the, like, original columns here. So because of that, um, we would need to use the columns for a as opposed to the columns of being So Yeah. So that&#x27;s kind of like the rationale behind. Why that? Why that ISS? Yeah. So s it looks like the first statements. False second statement. True, They&#x27;re true. And then the last two are going to be false.

James Chok · Answer

Okay, so this question will be answering with a serious off a tree and forces. So it was a if b is a basis for subspace hated in each. Better in hate can be ridden in only one way as linear combination of factors in B. This fact is true, which is, quite literally, the definition of a basis. And if you need more help on this question, if you go to the text book page 1 55 the beginning off section to 2.9 on the second paragraph, it says, is basically verbatim. Question be if B is a basis for suspicion of hate in the correspondence, makes hatred look and act the same as our p. So this is also true, which is literally told bourbon in page 1 57 See the dimensions off mo off a number of variables in the equation X because is there this is false is equal to the number off free variables in a executed the dimension off The column Space off A is rank A. This is true because this is that that this is the definition off, Frank. Hey, if head is a p dimensional subspace, there linearly independent set off the vectors in hedge is a basis for Paige. This is true by the base system.

Amy Jiang · Answer

Okay, so a is false because this is just the absolute value to determine it. He is also false cause determinants of a is equal to the determinant of 80. We notice from my term. Uh, but see, it&#x27;s joke. Murthy is false.

Doruk Isik · Answer

in this problem. We&#x27;re given a few statements and were asked for work. It&#x27;s that where it is true or false. So the first statement, uh, schools, Because the set of backers not only should be linearly dependent, it should also coincide which age? Hey, you could check the definition off base base. Second statement is true, and I&#x27;m gonna report you refer to the, uh, spending said here. Now, the statement is statement, you see, is also true. And this time I would refer to the section named two years off a basis. Uh, Indy, begin the statement. This falls because this standard method of producing a spend settle now a is actually always Cordish iss a re nearly Bennett sent. And it actually always produces a basis for no hey and less Damon. ISS also pulls because we need Thio look at D or use the different elements or give its Collins off matrix A not the magics B, which is the Rebecca Formal, eh? Because some columns of B mike nights be in the column space. Oh, okay. So we should also we should always use the pivot columns off our orginal matrix

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Suppose the solutions of a homogeneous system of …

Problem 18 Hard Difficulty

Answer

A. false
B. false
C. true
D. true
E. true

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A. falseB. falseC. trueD. trueE. true

Linear Algebra and Its Applications

Caleb E.

Samuel H.

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

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Samuel H.

Michael J.

Joseph L.

A. false
B. false
C. true
D. true
E. true

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

Linear Algebra and Its Applications