in-exercises-27-and-28-view-vectors-in-mathbbrn-as-n-times-1-matrices-for-mathbfu-and-mathbfv-in-m-2

Question

In Exercises 27 and $28,$ view vectors in $\mathbb{R}^{n}$ as $n 	imes 1$ matrices. For $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{n}$ , the matrix product $\mathbf{u}^{T} \mathbf{v}$ is a $1 	imes 1$ matrix, called the scalar product, or inner product, of $\mathbf{u}$ and $\mathbf{v}$ . It is usually written as a single real number without brackets. The matrix product uv $^{T}$ is an $n 	imes n$ matrix, called the outer product of $\mathbf{u}$ and $\mathbf{v} .$ The products $\mathbf{u}^{T} \mathbf{v}$ and $\mathbf{u} \mathbf{v}^{T}$ will appear later in the text.
If $\mathbf{u}$ and $\mathbf{v}$ are in $\mathbb{R}^{n},$ how are $\mathbf{u}^{T} \mathbf{v}$ and $\mathbf{v}^{T} \mathbf{u}$ related? How are $\mathbf{u} \mathbf{v}^{T}$ and $\mathbf{v} \mathbf{u}^{T}$ related?

Lucas Finney · Accepted Answer

So for this problem, we have two vectors U and V that are in our So we are treating them actually as u N v r mhm en by one dimensional nature seats. So they have en rose one caller and we want to answer How are U transpose V and V transpose You related where u transpose V is referred to as inner product And how are you ve transpose and the u transposed or transpose related where doing the UV transpose is referred to as the outer product. So if we consider our transposition rules, we didn&#x27;t know that the actually, I&#x27;ll start. That&#x27;s one second here, so u transpose the is equal to u transpose times v transpose transposed because if we transposed twice, we get what we started, which means that u transpose V. Now what we can do here is that u transpose the transpose transposed that is the same thing as the transpose times you all transposed using the fact that if we have a transpose be, or actually if we have generally you have a b transpose, then we get be transposed a transports. So we get that u transpose V is equal to the transpose of the transpose you. But since u and V our end by one matrices u transpose v is going to be Yeah, well, I&#x27;m not going to say it&#x27;s going to equal, but you transpose the is a one by one metrics, which means that obviously, since it&#x27;s a one by one matrix, you know it&#x27;s actually a scaler u transpose v transposed. If you transpose a scaler, you just get scaler back so similarly would get that the transpose you transposed is equal to v transposed to you. So that means that new transpose V is equal to the transpose you. The order doesn&#x27;t matter then, for the outer products. Well, you ve transposed. We can&#x27;t do that same trick if they&#x27;re in r N u v transpose is going to give us an n by n matrix. But what we can do you can say that you ve transpose is equal to you train or you transpose transpose and v transpose, which is equal thio v transpose, um or sorry not feature exposed v times you transpose transposed. So this is the most clear relationship that we can get for the outer product, that the outer products are the transpose of each other

Michael Jacobsen · Answer

in this video, we have two vectors were considering you envy and what we&#x27;re going to be doing with them is considering various products that we could form. The first thing we&#x27;re going to multiply is going to be u transpose times V. So first, let&#x27;s take the transpose of you. You comes as a single column, and so it&#x27;s tramps. It&#x27;s transposed will be a row. The entries will be negative 23 and negative for then I&#x27;ll copy envy, which is a B and C notice first before we multiply. That you transpose is of size one by three. The vector V is of size three by one. And because these dimensions here, which are highlighted or equal this is well defined and by the outer dimensions, the result is of size one by one. But that&#x27;s just going to be a scaler in this case, so we&#x27;ll take first negative two times the entry a. Then we add to it the second entries from the row and the column, So take three times be and likewise take native four times see, and this is our result of u transpose times V. This results in a scaler in our and this is sometimes called an inner product. Let&#x27;s switch this up. Next. Suppose we calculated v transpose times you What would we get in this case? Well, one thing we can note is that be transposed times. You could also be written in the form of U transpose times v transpose. The reason this works is a transpose operator would be taking the transpose of the entries in reverse order that would give us back the transpose and u transpose transposed is back to you. We&#x27;ve just calculated u transpose times v here and the result was a scaler. But if we transpose a scaler, we get back the same result. So v transposed times you is still negative. Two way plus three B minus four times. See, we know now that changing the order in these kinds of products of which gets the transposed does not affect the overall product. Now let&#x27;s change us up in a very big way and consider the next product which is going to be U times v transpose. Really. The only difference here is that the transpose operator came first in these products. Now it comes last. Let&#x27;s see what the difference will be for that. First, let me copy in you. It is negative two three and negative four. We&#x27;re multiplying it by V transposed. So that will be a B C. Written as a row vector. Now it&#x27;s do analysis on these dimensions. You is of size three by one, and be transposed is of size one by three. The inner dimensions match as before, so this is well defined, but notice something interesting here. The outer dimensions are three by three, and that tells us that the product is going to be a three by three matrix this time. So let&#x27;s begin by taking the product what we do when we take a Matrix Times another matrix Whether they&#x27;re considered Colin, Victor or row vectors is we begin to build the first column by using the very first column of the Second Matrix in this product. So we&#x27;ll go row by row, start by taking negative two times a That&#x27;s going to be our first entry, then take three times to say that&#x27;s our second entry and finally take negative four times a for our last entry. Then we go over to the second column. We&#x27;ll take a negative two times be. Let&#x27;s make this matrix a little bit bigger. There we go. Everything should fit in there now. Next, Let&#x27;s take three times be and finally negative four times B. So the pattern here is these numbers are dis multiplying. Be and producing this column here so likewise. When we go to the last column C, the first entry is negative. Two time. See, then we get three times, see and finally negative four times. See? And this is our result. This result is in our three by three, because this is a three by three matrix as opposed to being in our like we saw where the transposed came first and this particular product is sometimes referred to as an outer product. Now let&#x27;s try something similar, but still different. We want to calculate next v times you transpose, but we can use a specific trick just like we did up above here. If we change the order of our victors and interchange, which one has a transposed will have you v transpose transpose. But now we know that the transpose of this group would be first v transposed, transposed, and that would go here that next in the opposite order. Take you transpose So it goes over here. So we&#x27;ve already calculated you tens v transposed, and that tells us that the result is going to be equal to the following take negative two a. Three a negative for a negative to be three. Be negative for B negative. Two c three c negative foresee And what we&#x27;re doing is transposing that matrix. So I&#x27;m going to start with the first row and make it a column. It&#x27;ll be negative. Two way negative toupee and negative to see. Now go to the second row. Make this the second column. It&#x27;ll be three a three b three c and lastly, go to the third row and make it the third column so that we have negative for a negative for B and negative foresee for our third column, then this is the result of changing the order here on V. U transpose from UV transposed. We get essentially the same result, which is just off by the transposed

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answers. A question number seven from chapter 9.3 So split up into four different in the first part part, eh? We have to show that if you and be a each end by one column vectors that the matrix product you transpose B is the same as the doctor looked. You don&#x27;t be okay. So ah, um, to write you envy in a general way, we could write the components. It&#x27;s just general. Ah, And by warned column factors. Then let&#x27;s just figure out what the components off u transpose Hey, on dde you don&#x27;t be will be so u transpose v. Okay, first off, what u transpose? Well, it&#x27;s just the road back diversion. That&#x27;s so it&#x27;s got you go on. This first component onto your hand is the last one. That is just what we got rid of both. Just be one to be on. They were moved by these two like we would ah vectors. So we&#x27;ve got, um first up, we got you one times V one that would add to that, uh, you two times. Me too. On all the way up Thio U N times via. That&#x27;s just working out. What? Uh, this, uh, comb that. That this vector, uh, the product of the vector u transpose. And for years, we have a look at this and think What is that? Well, that is just the definition off the dot product. You don&#x27;t. They s So that is what we defined by you don&#x27;t They s so yes, it is true that u transpose J is equal to you, Doc. Next up, part B. So we&#x27;re asked. We&#x27;re told that V is a three by one column Vector given here on a is a three by three. Make Thanksgiving here. So it has to show that the transpose of you times be sorry. Transpose of a Times B is the same as three transpose times a transport. Okay, so let&#x27;s work out. What a times. The year&#x27;s first. Let&#x27;s just write them out. So a is this matrix? The is the column Vector to three five. So first I forgot at one by two, which is to you Plus to buy three, which is six close on my five. Just five. They&#x27;ve got one by to you, which is 23 by three, which is nine onto you by five, which is 10 one by two, is two zero by 301 by five. It&#x27;s five. That&#x27;s good. Equal t 13. 2195 7 That&#x27;s what TV is. Now let&#x27;s work out. Well, v transpose a transfers is okay, so V transpose is the row Vector 235 on a transpose is where all the collins become Rose. Then we could just work out what this is. So the first component is ah, two times one, both three times two plus five by one. So that&#x27;s two plus six plus five, which is 30. The second component is two times one just two plus three times three, which is now in plus two five by two, which is 10. That&#x27;s that&#x27;s gonna be 21 on dhe. Next up, we got two times one, which is two plus three times the average is zero plus five by one, which is five a two plus by which is seven. And now you see that this is just the transfer&#x27;s off. This therefore a the chance Bos it&#x27;s just people to be transposed a transpose apart. See thruster. Show this in the general case. Okay, so let&#x27;s let a be a general and by m Matrix, So components we could write a general way. So this one should be a 11 that we&#x27;re gonna have better a 12 a 134 weapon, but the top all the way to a one. And on down the first column been of a 11 81 831 all the way to a and one. These were gonna continue along to fill it with matrix until we have in the bomb, Right? A and and V is a general. And by one vector so we&#x27;re good, right? That&#x27;s just be one all of it to be And okay, so first up, let&#x27;s find out what a times he is. So it&#x27;s gonna be a column vector off shape and sorry of shape em, but I won&#x27;t. So, in the first component off the column vector Well, it&#x27;s gonna be a 11 times V one close. That&#x27;s gonna be a 12 times of e t A on three times three etcetera, all the way up Thio a one and times bia. Okay, then next up, it&#x27;s gonna be a 21 Time to be one plus a +31 That&#x27;s a 22 times. Me, too, was 83 times beaten at that time. Speedway it all the way up to a to end times be yet We do this all the way down until we get to a em one times be one plus all the way up to a and and times via So that&#x27;s what I&#x27;ll better, uh, defined by the multiple of matrix A buy back to B. C. Okay, so now what about V transpose times A trans boats V transpose is just the row vector Do you want to be? And a transpose is just a where all the rows of columns and all the columns of Rose this is all the way out m 1 a.m. one all the way down to a well and all the way to a M. And Okay, so what&#x27;s this? Well, this is gonna be a row vector. The first component is gonna be the one times a one that&#x27;s a woman, uh, close. Be two times they want to plus the end times they want. Okay, I&#x27;m gonna have to do that for each of the other components. So we get to the last component in the robe actor, which is gonna be the one time&#x27;s a em one close all the way up. Thio The end time&#x27;s A Mmm. Okay, so all this is is just the transfers of this, and vice versa. So therefore, we can say this It is just or rather, who said this Ah v transpose a trance Berries. He&#x27;s just gonna be the transpose off the one above, which is a The transfers that completes the proof for part seek left party were asked if I at rusty show with this holds for ah, instead of using A and V. What these victories and A and B would be is a matrix. So for this reason makes sense, then be needs to be on end by M matrix. So we can write that in general form where the components are be 11 all the way up t be one, um, all the way down to be Enron on all the way down to be and and then we just think of this as just a collection of column backed is call each of them a safety one be to etcetera all the way up to you, B m that makes each of these make sense like you&#x27;re gonna get Answer when you must put them together. Um ah. Nde. We&#x27;ve shown that it&#x27;s true when b is just a single column vector. So if we make B a a matrix made up of the common factors, then it&#x27;s gonna be the same. And then it&#x27;s gonna be true for all of them, right, Because all a times B is doing is keeping you always just keeps the the columns of B the same you always most Flying Age column and would be five rows of a on. Same with this. So therefore, since it&#x27;s true for part C, then it must also be true. Hey, Pop.

Chris Trentman · Answer

were introduced to the singular value decomposition method of matrix factory ization. So suppose that a is equal to u times d tend to be transposed where u and v r n by n matrices with the property that you transpose u equals I and that v transpose v equals I. And where d is a diagonal matrix with positive entries. Sigma one through sigma n on the diagonal were asked to show that the matrix is in vertebral and to find a formula for a inverse. Now we have that both u and V transpose are square. So the conditions on U and V imply that you and the transpose are both in vertical matrices by the in vertebral matrix. The&#x27;re, um so it follows that you inverse equals u transpose and we have that be transposed in verse is simply ve Now The diagonal entries of D are non zero. Since none of these entries or zero it follows that D is an in vertebral matrix, and we have that the inverse of D. It&#x27;s a diagnose metrics as well civically with entries Sigma one in verse through sigma and in verse on the diagonal. So it follows that a is a product of convertible matrices. So it follows that a itself is in vertebral. Not only is a convertible we have the a inverse is the inverse of you d be transposed and this is going to be be transposed in verse The inverse universe which is equal to the de in verse um, verse and these air Easy to our Sorry you transpose since you universe is u transpose and these air easy to compute

Elham Kordzadeh · Answer

a because 12 three full mark Chris be equals two. Fine, I think seven. Hate a plus B. He wants to Sikhs age and one a nos the once a bad boy a les bee Because to speaks Kate 10 Twinned wanted hurt boy Weeks aid And Michelle wants to 100 16 140 four 180 200 20 A tube because toe a continent by a because to one to three wanted employ bon to three Oh, we should Waas Who? Swimming 15 My knee beat. Who equals being no simple boy Be because to fine speaks seven eight months Depends bar Why I think And then Kate we should Waas Who? A 60 seven 70 eight 90 Bon 100 on six Tour A The it wants to two months dead boy one, three whole One super boy While in there Thanks. I live in eight equals. There&#x27;s you need worth e Oh, 100 on 80. Thanks. Then we have the can condition off Okay too. I lost to a be yes beach. Who? A horse living and 15 22 No sauce nursey eight. 44 86 100 Los 60 seven 70 Aid. 90 Come on. 100 six. Michigan wants to 120 132. The 100 is 92 200 28 and then they have 80 equals two 12 Screen or multi brand boy flying Cease seven eight equals two, 19. 20 to four c three on to be Hey, it holds, too Five weeks sitting. Eight 21 Screen Hold Michigan calls to 23 30 Pool there See Von on 46 on. Then they have the calculation off. 80 to a 21 loss. Any to plus Gaby last be a nos Be true. Equals from the table. Big camp a little off the most places becomes 160 144 182 100 When E.

Ashley Boni · Answer

so we won&#x27;t show that he is a linear transformation by finding a matrix, um, such that when we multiply it by vector X with four components, we get this as an output. So this transformation is mapping a vector in our four to vector or well, just the number in are a real number because this is just a number. If we plug in ex one next three x for so we know we need a matrix to multiply x one x two x today, that&#x27;s four. So that means we need to have four different columns one to multiply each of these, uh, vectors. But it&#x27;s being mapped to just are so we only need one room so we can stop. So we need four different components that multiply at claw on x two x three x for and we need it to give us to x one plus three x three minus or explore. So the coefficient for X one is two coefficient. Rex to zero coefficient for X three is three and the coefficient we need on export is negative. For so ignoring these lines. This is our matrix that we need to multiply

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Linear Algebra and Its Applications

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Alayna H.

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Absolute Value - Example 1

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

Alayna H.

Caleb E.

Maria G.

Kristen K.

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

Linear Algebra and Its Applications