rank-1-matrices-are-important-in-some-computer-algorithms-and-several-theoretical-contexts-including

Question

Rank 1 matrices are important in some computer algorithms and several theoretical contexts, including the singular value decomposition in Chapter $7 .$ It can be shown that an $m 	imes n$ matrix $A$
has rank 1 if and only if it is an outer product; that is, $A=\mathbf{u v}^{T}$ for some $\mathbf{u}$ in $\mathbb{R}^{m}$ and $\mathbf{v}$ in $\mathbb{R}^{n} .$ Exercises $31-33$ suggest why this property is true.
Verify that rank $\mathbf{u v}^{T} \leq 1$ if $\mathbf{u}=\left[\begin{array}{r}{2} \ {-3} \ {5}\end{array}ight]$ and $\mathbf{v}=\left[\begin{array}{l}{a} \ {b} \ {c}\end{array}ight]$

Subham Jyoti Mishra · Accepted Answer

question you is record toe the tree I these core toe Yeah, the PC. You will be the crumbs for the physical to achieve by but other air B c Very good. Minus to And we have toe verify their Harang off. Your ability is less than a quarter of one now. I&#x27;m not without us. All these metrics Motive, Nicoson. Some of the voting who air could be conceived. Speaking mhm, Casey by air by E. C. They introduced this metrics using a little prisons, first of all, performing, uh, from okay to promise by our So, yes, there&#x27;d be you see? And And we hope that we can call them. They are the biggest re are you Get in Seattle. 0000 He didn&#x27;t simply fight for other. So we get a three c 00000 now for uH huh P c. North Dakota zero maximum rank off a secluded one, the division of the economy and more mature rating every year. Zero then, frankly, zero. Yes, the little one

Runpeng Li · Answer

you too. So work even Victor 12 And we need to find a factor. VII such that new times. The transposed is a two by three matrix, which is one collective three for and two to negative six and eight. So one thing you notice that after you is actually a two by one matrix. And here the Matrix is a two by three matrix. So the only way we can you can find a such a vector. V is a one by three. Well, actually, Dr B should be Ah, iss one by three Vector Well, sorry. Three by one, three by one vector. And that way the V transpose this issue. So the U transpose Sorry how we writers be transposed we&#x27;ll be one by three So the modification of these two vectors will be two by three. So that&#x27;s our matrix here. So what is the vector? U transpose Well, first we have the first rule of you. That&#x27;s one we have one time. Something that should be so I offer It&#x27;s right down the transports, our vector Victor B to B A B c. So we will have something that one times want one times a to be equal to one. So the only way to make it possible that it&#x27;s one. So we have something 11 times is something that is negative. Three. So you want me to make it happen that in negative three and so on so forth. We haven&#x27;t four or a C, and that way we can. Well, actually, you get the tube. I cremate matrix like Like what we require here. Because 12 that&#x27;s the one. Times one is 11 times selected. Three. It&#x27;s three. One times four is four and two times one is 22 times taken. Three. Think of six. Two times four is eight. So this is our transport, something Specter. And in terms off, better be. It&#x27;s just a three by one three by one matrix, that is one nectar 34

Subham Jyoti Mishra · Answer

in this question. We have American Tory&#x27;s. I just grew by three mattress and who is the most? Call them off the that has You want you to end you three Parnham&#x27;s You&#x27;ve been given their rank off here is you got one. It implies when rank is one you want is the pieces the column space here It implies you to learn you three dependent you want so they can be a scandal. Our X And why that&#x27;s that your political X, you one. You&#x27;re three years ago. Oh, boy, you want? So in this case they will become human ex You one. Why, you one This can be returned. Us. You want you to your tree. I get tired one. Thanks. Why? And is this you, Victor and his The transport. So we have a vector V, which is one X? Why such that it in middle Tennesse? Yeah. So you have a camera is very nice. You wonder certain place is a you won. The transpose of them is The key is when you want is not a zero. You one is a gold zero And you go where become unique and you feel become, Sir, you to there appears be there different 01 is there And Abel in your school we transfers Then do you want you to go 34 0 and you re will only be present in a very require your tree? Review the prosperous Be very procedural, you know one. So in any of the cases he can be expressed us you. I think they&#x27;re very good transport so that I existed Vector space be

Sriram Soundarrajan · Answer

Hello? Did you hear? The given metrics is minus 2403 Fire about it. That little little and high three equals my little little 00 Seattle&#x27;s it&#x27;ll We need to fight, I think because it God full 10001 little there are there are in the days, minus school for their own The fire that so it comes to be my best to formal cathedral three by night, 00

Joe Lesueur · Answer

the following problem tells us that matrix A is m by n matrix Matrix B is a p by n matrix and we must prove that a times the transpose of be all transposed equals matrix b times the transpose of matrix A. To start this problem, we will use index notation. So a times that trans, whose of matrix b all transposed is I&#x27;m Jay Using that notation, we can take away the final transpose of the entire major cities, have a times transpose a matrix B and we can just foot those and put J i since we&#x27;re transposing the entire operation. And this is also equal to the summation cake. Want em of a J K be transpose Okay, I right now these case are just placeholders were mainly focusing on the the I N J notation. You can see a transfers to just the JK notation all the transpose of B to the K nine notation. I&#x27;ll find them together. The case with cancelling you&#x27;d have a B j I. So the next step of this problem would be to take away the transposed and move it to a We can set this equal to summation K equals 1 10 of be I today. We&#x27;re just taking that transpose here and applying it by switching these around and this is multiplied by a okay, J transposed. We&#x27;re doing the opposite here, or that gene K now flips because we&#x27;re adding the transpose operation. And now, since this notation can also be written back in the form of matrix be multiplied by the transpose of matrix

Nez Nikoo · Answer

okay. We want toe. Um, prove that Property nine, which is determiner of inverse of AIDS. You go to one over, determined off a itself. In order to do that, we can say that a matrix a times is inverse. We know that is equal to identity matrix. Okay, now we have to make sure that this is in versatile matrix because we do not want determine a to b zero and it&#x27;s and my end. Okay, So in order to prove this, what we can do is we can get the determinant. What we want to prove is is I&#x27;m not going to write it down, but it&#x27;s basically DT A a minus one is equal to one over DT A. So what I&#x27;m gonna do is I&#x27;m gonna get a determined on both sides. So what I&#x27;m gonna do is I&#x27;m going pretty a determine it of a in verse of a equal to oops you call to determine it. My we know determine of I is one. Okay. This we can distribute the determining if that makes any sense. I&#x27;m just going to basically separate this determined a times determine it, love a in verse. So by looking at this. In order for me to get a number one, these two have to be inverted of each other. Or what I can do is basically since determinant of a It&#x27;s not Zero is in versatile. Okay, this is a key thing. You have to make sure this is the key, right? This right here, then I can divide both sections. Both sides. This side on this side, both determinant of a Okay. If you do that, then you&#x27;re basically canceling this. And we proved that determine it in verse of a is equal to one over determinant of a That&#x27;s it. Thank you.

Problem

Rank 1 matrices are important in some computer al…

Problem 31 Easy Difficulty

Answer

$\operatorname{rank} \mathbf{u v}^{T}$

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$\operatorname{rank} \mathbf{u v}^{T}$

Linear Algebra and Its Applications

Lily A.

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Anna Marie V.

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Vectors Intro

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

Lily A.

Catherine R.

Anna Marie V.

Heather Z.

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

Linear Algebra and Its Applications