suppose-the-factorization-below-is-an-svd-of-a-matrix-a-with-the-entries-in-u-and-v-rounded-to-two-d

Question

Suppose the factorization below is an SVD of a matrix $A$ , with the entries in $U$ and $V$ rounded to two decimal places.
$A=\left[\begin{array}{rrr}{.40} & {-.78} & {.47} \ {.37} & {-33} & {-.87} \ {-.84} & {-.52} & {-.16}\end{array}ight]\left[\begin{array}{ccc}{7.10} & {0} & {0} \ {0} & {3.10} & {0} \ {0} & {0} & {0}\end{array}ight]$
$	imes\left[\begin{array}{rrr}{.30} & {-.51} & {-.81} \ {.76} & {.64} & {-.12} \ {.58} & {-.58} & {.58}\end{array}ight]$
a. What is the rank of $A$ ?
b. Use this decomposition of $A,$ with no calculations, to write a basis for $\mathrm{Col} A$ and a basis for Nul $A .[	ext { Hint: First }$ write the columns of $V . ]$

Amy Jiang · Accepted Answer

Okay, so we&#x27;re given negative Jews or one so one job. I never want to. Joe times 50 Joe or jail cells over here. And zero, 21 So one so and 14 You did, too. Product. This is calling. He so we have our Are you in value one. I was like that one. So I am values. I need values on the diagonal, right to have a family of five by then four and in our X one is equal to that&#x27;s later to 01 Next to is equal to it&#x27;s just call a mechanic with a 10 an X three is equal to negative one, too.

Runpeng Li · Answer

all right, so in this problem work, even the matrix A and row equivalent being tricks be. So the first thing we need to do is to without calculation, to lease the run K and dimension of No A. So the run k. It&#x27;s just Ah, we just need to count on non zero rose reduced in a row, reduce toe matrix. Speak. So that is actually three nuns a year old rose. So the next thing you mentioned up, no way it&#x27;s to count. Yeah, and then about three up three barrels. So, in this case, X five, he&#x27;s always zero. If we consider the axe equal zero, the equation being very question and x $5 is always zero and ext ext. Three will be determined by export an X five but x 50 So experts three will be. We only be determined by its export, so explore it&#x27;s a pre bearable now x one will be determined by x two x three x for an X five. But on Lee, X two can be re bearable because x three will be determined by explore an X 50 so x two It&#x27;s a three variable so the dimension turns out to be too. Okay, so the next thing is to find the basis for column A or column may the basis. Uh, the only thing we need to do is to find out people people columns, and write down the people of columns in himself for entries up eight. So, by observing the matrix B, we find that people columns are come one Come three column five. So we just write down the first column there column and the fifth column off, eh? So that is one negative, too. Neck of three and three and four. Um, we have neck of six of six and four. The fifth column is nine negative. 10 and next three And there. So we&#x27;re done. Okay, next thing is to find Roadway. So in order to find the row A we still we need to observe the reduced. A rueful reduced to roll matrix, um, to observe. Dad, we just write down the rose that are not are not all zeros. So in this case, we need to exclude last row, we keep the the rest of the rose. So then we&#x27;ll be so first factor. Yes, One back up three 057 Second ISS Neural zero two, two. Sorry to negative. Three eight they call him is 0000 Okay, so the next thing is to find the basis for no way. Well, to find that now you need to do we need to do further. So we have to turn this, uh, Roy Quinlan to be relatively equivalent matrix B to, uh, reduced ancient form. So our first right down be one naked 30 by negative seven. And they&#x27;re all No. Two connective seven, a neck chemistry and eight Sorry. Hate. And they&#x27;re on their own. Zero. There are five and all deals for the last room. Okay, to do the washing elimination, where youse I say we first divide the second roll by too. So that gives 1230 by next of seven. You 01 naked, three have, and four. And we also divide by divide a 30 year old by five so that it will be the rose 0001 and all the roads for the last. Let&#x27;s roll. Okay. No. We use the second roll minus four times out four times off the third girl. So that is one negative. 30 by seven and 001 naked. Three have and zoo and last row. Sorry. 1/3 row is 00001 And all the roads for the last roll. So it needs to selling reduce station form. Okay. Um Well, the next thing is to find out, find out the free variable cleans. So that is It&#x27;s five zero on next three is we have x four and x one is real text too. Minus five x four minus seven x five well X 50 So he&#x27;s just a three x two minus five. Explore. All right, so right here we have, we have, actually, that is X three, just 3/2 off x four and x one is reacts to minus five X for Okay. So in order to find out basis, remember, we have to have to free parables here so that the sex to our next four. So the way to do that, we first have, um, check. X one will be three f x two minus by export and next to would be one zero and x three is zero x two, but three have off x four and next four is just zero and one. So our faces hey says is so by our calculation is for three. Three. Sorry, we 100 and 9 to 5. Do three have on the one. All right.

Arsalan Afkhami · Answer

we need to find the, um, change of basis metrics from B to C to do that first were right the first victor off, um, with that basis be, which is negative seven for four, very right call to. So we need to find a vector with three components. C one C two c three that when multiplied by the victors off or the basis see gives us being or Victor I returned. So what you do right? See? One supplied by the very specter warned 10 Let&#x27;s see two. He&#x27;s 01 bone and then C three buys three native warning fun Then here you have X. Why the X is a call to see one conclude by, except the first specter sitting with fire by except the second victor and see three multiply by X up the third victor and similarly for wines that so I write. This isn equation, which you see. One plus c three c three close to 97 c one plus C two, maybe minus C three each call to four and then see too minus C. Three years a close to four. Just we can rewrite this one. C two is a coastal for plus C three and replace it. Um, here then we have This is not a provisional to our questions. When we consult with easily and gets he won&#x27;t seek to see very close to and ah zero All right, on negative eso the first victor political basis. See, he&#x27;s zero so extended. So first we find the first column off the metrics we are looking for. So you do the same for the second victor or the basis Be sore right here. The second victory for four to limit Juan A similarly be right See one by 110 plus C two supplied by 01 plus c three expired rise 381 on one. I write their stuff equations here. Yeah, I see one Thus reseach three, composed to four she won plus C two minus 63 years goes to to on C two plus Esser needy c three He close to anyone you can simplify here say see too close to one plus c three and then replace and solve for C one, C two and C three find B Victor here the second victor related to see which is going to be, um three. You have to said on What&#x27;s it? So for this victor, we do the same. Write it down. You have seven by Geo because to see one described by one and zero. Let&#x27;s see too easy. Close to zero plus C three. He&#x27;s no one, right? This is no accordionist year. We have C one plus three. C three goes from me three She won plus C to minus two C three. He&#x27;s a close to five and fancy too. Minus C three is a cost to zero easy. So you have to see to Izabal 63. Just place it here and then you have, as we start compressions here that he s so fine C one and C two C three. And he said Victor would be by before, before at thes three letters urging three columns off the the mattress if you&#x27;re looking for. So as these are the columns off the metrics rights, my P from B two C, easy close to zero. And men, this is multiplied by, um, the victors off. Yeah, basis B is going to give us basis. See genetic to the basis. See, this is the right now

Ernest Castorena · Answer

I like it for this question. They want me to determine that I mentioned of the North Space Given Matrix A In this case, they&#x27;ve given us one negative 236 What can we do? Well, we can actually just compete the rank of the matrix of thinking, cause we know that the ring of a plus that I mentioned of the North Base is equal to the dimension of the code. I mean, that&#x27;s the rank Naledi them yet Well, looking at our matrix we actually see of Ah, this is just the first column multiplied by three so we could eliminate actually ended on the Swiss 13 00 And we&#x27;ll from here it&#x27;s pretty clear that the rank of the matrix is just one. And while the co domain this is the two by two matrix is they mentioned to the one plus that I mentioned were no space. It&#x27;s too, so we can subtract one from both sides and we know that the dimension that we want is one

Subham Jyoti Mishra · Answer

does that he is. It will go. This is a four by six metrics before by five records. Who? 93. Six? Who? My When it&#x27;s too three on 3. 93 94? Uh huh. Minus six nine. Bye. Nine. When it&#x27;s too mhm. Mhm. Venice for one. Mhm. Let&#x27;s go. This mhm. This has been reduced toe. Wow. A cloned home, Pierre, which is too honest? Three 6 to 5 00 three minutes. One money, 000 We won three. Mhm. That way. So here we have to find out in the ankle. Yeah, the physical, the diamonds or Koroma. Yeah. So this is you go to the number all non zero. Uh huh. Bruise little markets. So if you have one, you go on a tree. The rank history, the diamonds, you know? No, none. Not just a physically brutal and minus drank off a Which is equal. And his number all problems, which is five minus three physical, too. Now the pace is called or loans notice here the equal then values off column vector corresponding to By what elements way have. These are the private elements, So call on one, three and four so corresponding Tothis all the metrics. We have the vectors two minus two or minus two that we have six minutes. 393 And we have proved ministry by managed for Then you can include the basis. Well, Metro north tricks. So this is really cool. The royal elements corresponding to non zero rose. So you shall be guru buh minus to be Mm. Right. Um, 00 three minutes. 11 00 zero one She So it is the basis of electric said No. This is off bottle metrics off here will be equal. So listen Vector X corresponding to see X is equal. Zero essay is the reviews for a week long form of the metrics. You see, in this case, the acquittal, the one the most three upon to 0 to 32 Little little 11 day three. I&#x27;m going through. 000 13 00000 So corresponding, Solis. And more tricks. Uh huh. Three by two x two. That&#x27;s nine by two Ex life. And we have a school here and we have minus four x i, the country for the next three. And then we have minus three x five And, uh, X bye is always no tricks. So therefore, the base he saw No mhm er the brutal he trying to hum his heroes. Heroes, heroes. No. Um Mhm nine by school zero minus four by three. Mhm minus three in the morning. Mhm. Looks like these are the Yeah.

Problem

Repeat Exercise 15 for the following SVD of a $3 …

Problem 15 Easy Difficulty

Answer

Note that $A=U \Sigma V^{T},$ so matrix $V$ in this decomposition is given trasposed,
so $v_{3}$ is not last column but last row of given matrix.

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

Linear Algebra and Its Applications

Catherine R.

Alayna H.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

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Note that $A=U \Sigma V^{T},$ so matrix $V$ in this decomposition is given trasposed,so $v_{3}$ is not last column but last row of given matrix.

Linear Algebra and Its Applications

Catherine R.

Alayna H.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

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

Catherine R.

Alayna H.

Caleb E.

Maria G.

Note that $A=U \Sigma V^{T},$ so matrix $V$ in this decomposition is given trasposed,
so $v_{3}$ is not last column but last row of given matrix.

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

Linear Algebra and Its Applications