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Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\}$ and $\mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}\right\}$ be bases for $\mathbb{R}^{2} .$ In each exercise, find the change-of-coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$ and the change-of-coordinates matrix from $\mathcal{C}$ to $\mathcal{B} .$$\mathbf{b}_{1}=\left[\begin{array}{r}{7} \\ {-2}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{2} \\ {-1}\end{array}\right], \mathbf{c}_{1}=\left[\begin{array}{l}{4} \\ {1}\end{array}\right], \mathbf{c}_{2}=\left[\begin{array}{l}{5} \\ {2}\end{array}\right]$

$P_{B-C}=\left[\begin{array}{cc}{2} & {3} \\ {-5} & {-8}\end{array}\right]$

Calculus 3

Chapter 4

Vector Spaces

Section 7

Change of Basis

Vectors

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get. So in this problem, we present you to find the changeup Coordinate matrix from you did see so you to the scene. And this will be this can be achieved by first food. The column vectors of be on right heart, its matrix seven nectar to two active one and column column vectors of sea or one. And why two? So the way to do that we do the, um, reduction. So we first divided first roll by poor 5/4 7/4 and to quitters. And second of all, would still be one too negative. Two and one next one. Okay, so this gives. So we used the second row diva minus the first roll. We have you to minus 4/4. That is 3/4 and 7/4 minus 7/4 minus two. That is 1/4. Well, yes. So he's actually we used one minus one and two minus five over four. That's 3/4. And next to minus 7/4. That is negative. 15 over four. Right, Because selective to minus. Just for 60 minus 7/4 that's Ah, eight rito plus seven. I could sit 15 over four. Yeah, and the last one is 91 minus two or four. So that is. See, um, keep six. 04 And we still have the first roll. Libel before seven over four and tour. Okay, now we first to turn the second row. We multiply second row by for over three so we can turn this entry to be one. So we have a life. Local traitors, 7/4 two porters and 01 and three over four. Yeah, it's one and or third times negative. 15. What? Hers that is 5 95 and negative. Six or four times where? Oh, where Three. That is connected to. Okay, So for the first rule, look, I use 5/4 off the second reel ad. First role. Sorry. Minus first, So we can cancel the entry of the entry here. So that's 10 and 544 times negative. Five used. Seven over four minus is value. So it's seven over four plus 25. She's 32 or four. So that's eight. And we also have to over four. Finest. I'm over four times negative too. So that is 10 plus two or four. So that's three. We have three here. 01 and active, applied negative too. So I were change. According the Matrix from B to C is eight three negative five and negative too. And the inverse C two B. It's the inverse off the you to see and remember how we find it. Find the inverse up two by two matrix First the best A determinant. Uh, this matrix, which is? Uh um you to see times we have the changes have to entertain you The diagnosis that selective too. And eight No, we have to take the negative 23 And what? So what's the determinant determined then? Days negative 16 plus 15. That's negative one. So multiply by multiplied by wants us to read negative five and activate.

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