11. 9 points) Let E = {V1, Vz, V3} = {(1,0.0) , (1,1,1), (0,0.1)} and F = {u1, Uz, Us} {(1,0,-1),(2,1,1), (0,0,1)} be bases for R3 Given the following matrices A = ...

Find the change-of-basis matrix $P_{C \leftarrow B}$ from the given ordered basis $B$ to the given ordered basis $C$ of the vector space $V.$ $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\{(2,-5,0),(3,0,5),(8,-2,-9)\} \\C=\{(1,-1,1),(2,0,1),(0,1,3)\} \end{array}.$$

This question is a little less computational. Lee Heavy. Ah, See the information that we want from the problem? They just want us to find the dimension of the north space of a given matrix. Okay, well, let's see if our matrix is a negative. 933 negative five. This is a one by two matrix. So it should be out putting five dimensional vector. However, we only have one entry to express leg. The possible solution. That means that five subtract one for therefore, um degrees, which are completely ignored by our matrix. So the null space will definitely be for and the reason why that helped. Because we knew the range waas one. And we know that if we had that I mentioned of the rain and the dimension of the hospice, we should get the total dimension of the cold. I mean,

To solve this problem, we can simply compute the inverse off the metrics up then in problem 19. But we can also solve and separately again. So first I write the first victor off basis. See which is one you want one. Then we need to find the victor, which, as the completeness she ran c two, c three that when it's multiplied by the victors off basis be gives us the this victor so called to see Juan fly by the first victor 50 plus C two, three year five plus C three, eight to nine on. Then you need to write this stuff equation. And so here is X. Why she wants by the X off the first victor. Plus she too. Bye bye. Except that second on c three, Why would accept that Victor is going to give us the X under the con side. So to see one plus three C to Los age C three is it called one? I was five. She wrong? Plus, we don't have the C two for the securities and so I only see three thanks to 36 seas three musical. So negative one, Uh, then five c two minus nine C three musicals. One here you get. We can't just simplify. I just want to say Ah see? Want easy call to, um one minus two C three. She lighted by five and then replace. See one here. Then we will have a system of equations. We, uh, two equations. And then again, we do the same. The substitute C two or C three. We have one equation. One, uh, solution. And then we can solve c one c two c three. Other solution would be to use Gaussian elimination. So here we will have an augmented metrics. You have to three eights. There is one and maybe five CEO. You too, Geo. If I maybe nine and here made one and one of what you need to do is to eliminate one side of thematic. So this is the they're gonna, for example, we need to most by the first row when number on then added to be said row as to So when it's added to the federal get zero on added to the second row. So this is zero. When we do this way, get, uh, she wants C two, C three and the answer, um, for but first equation that you wrote here. He's and make to her that we want you letting two bases b your cheese feasts. No face here. This was the first victor. You have two more rectors that we need to so find So a second he's to geo one. It's a call to see one to negative 50 as before, so it's exactly the same. See to 30505 I was three h negative to nine. So here is saying we have It's like the same subdivisions. The only difference here. So here we had to see one plus three c two plus 83 c two plus AIDS c three. No, it's a call to to and the thing in second month was for maybe five c Ron on this to see three. No, it's a call to zero instead on five. See to minus nine c three physical, 21 So assuming on TV solve its by simplifying or using busing and determination to find C one. C two c three maybe gets there you to that's to be or soon just on one. So this is the second victor and we have one more so and last form. You want three musical to see one the same as before 20 Last see two, 305 US C three, eight to nine. And here, your assistant of equations that we had above that you have C equations that will give us c one c two c three on They said Victor will be negative. 7 45th 25 So these three vectors that we have obtained are there three columns off the change of basis metrics. So it's injured. Metrics from sea to B is equal to fresh. Be right. The first victor now is the second victor that we found. I'm fair. This is the final answer off this question.

Get for this problem. They want us to find a basis or a subspace of square major cities by two rial entry. The subspace is all matrices with trays. Zero. What do those look like? Well, that's right Now and B. C Well, if the trace is going to be zero at this point, we have to have negative. Is this an injury? And now let's This is an arbitrary element of our space for any choice of ABC. But now let's write our main Chrissy in a different way. Negative. A 00 You can write as the some of these three mantra sees 000 seeing plus 0 +00 be Okay, Well, now we can throughout the scaler. No, since the one ce time 0001 plus spee times +0001 And so we see the this arbitrary element of us has been uniquely written in terms of these three major C and they do form a basis for our subspace. Okay, Awesome.

Yeah. Okay. Um so suppose were given a basis of two vectors, which is uh first factors 1 -1. The second vector is 1, 1. And uh were also given a W, which is another factor that is equal to 10 And I want to find the coordinate factor of W. With respect to this basis. Okay, How we do that? Well, let us reformulate this to an easier to handle form. So I'm going to collate the basis factors into this one matrix over here, and I'm going to multiply it by some arbitrary uh you know, like uh column vector. Let's call that X. Y. And then that's going to end up using us back the W which is 10 Okay, so finding the coordinate vector of W. With respect to this basis. Okay. Is the same as finding this column vector X. Y. Under this matrix multiplication. This this problem that I've set up over here. Okay, so how do we solve for X and Y? Well, We just rearrange the terms. Okay, that's equal to 1. -1. 1. The inverse of that Gay Times. 1, 0. Okay. How do we find the inverse of a two by two matrix? Well, again, let us recall that if I have a two by two matrix, A, B C. D. And I want to find its inverse. Well, that's going to be equal to one over the determinant of this matrix, which is a d minus Bc. Okay, and then for the actual matrix I need to exchange the diagonals so it's D. A. And then the off diagonal stays where they are. But I have to put a negative sign in front so negative b negative. See so this is the formula for finding the inverse of a two by two matrix. Okay, so over here I'm going to apply this formula. The determinant of this is one minus minus one. So it's just too, so it's one over to over here and then I have to exchange the diagnose which is just 11 Uh And then over here I have to put a negative side in front of the off dying. And also I have negative 11 Okay. And then I multiplied by the same column vector 10 Well, what is this going to give us, this is going to give us the vector 1/2, 1/2. And therefore we are concluding that the coordinate vector of W with respect to this basis is going to be one half, one half. Okay, so let's clear the screen and work on the second example. So I suppose this time the first basis vector is one minus one, and the second basis factor is 11 And the w. Okay, that we're given is 01 Okay well how would we go about finding the coordinate vector of W. With respect to this basis. While we do the same thing, we're going to collate the basis factors into this one big matrix, multiply it to the column vector X. Y. And set it equal to W. And to finally quote in that vector of W. With respect to the spaces. We just have to solve for X. Y. Well how do we do this? Well we have X. Y. Is equal to 1 -1, 1, 1, Inverse Times 01. And how do we do this? Well again we take the determinant, so 1 -1. So it's 1/2. Okay. And then what is that? Um how do we figure out the matrix part? Well the diagnose have to be exchanged but then they're both ones so they just stay, you know they just stay the same as 11 and they have to put negative signs in front of the off diagnose. So I have minus 11 like this And then we're going to have a zero one. Okay? So uh if you were to multiply this thing out, uh this picks out the second column of this matrix, so this just becomes minus one half in one half. Okay. So therefore we are concluding that minus one half one have this column vector is going to be the coordinate factor of W with respect to this basis uh that I've shown here on the left.