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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 a vector space $V,$ and suppose $\mathbf{b}_{1}=-\mathbf{c}_{1}+4 \mathbf{c}_{2}$ and $\mathbf{b}_{2}=5 \mathbf{c}_{1}-3 \mathbf{c}_{2}$a. Find the change-of-coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$ .b. Find $[\mathbf{x}]_{\mathcal{C}}$ for $\mathbf{x}=5 \mathbf{b}_{1}+3 \mathbf{b}_{2}$

A. $\underset{C \leftarrow B}{P}=\left[\begin{array}{cc}{-1} & {5} \\ {4} & {-3}\end{array}\right]$B. $[X]_{C}=P[X]_{B}=\left[\begin{array}{cc}{-1} & {5} \\ {4} & {-3}\end{array}\right]\left[\begin{array}{l}{5} \\ {3}\end{array}\right]=\left[\begin{array}{c}{10} \\ {11}\end{array}\right]$

Calculus 3

Chapter 4

Vector Spaces

Section 7

Change of Basis

Vectors

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call them too. So we need to find Thea Person Need to find the change of courting the matrix from B to C and your work even be one is elective C one us or I'll see to it. Sorry, just or I'll see to. And you too is five C one minus three. Upset you. So by over your own given in the 15 given in the chapter 4.7 we I know that the Matrix changeup quote in the Matrix is you want record C and B two bracket See? And this is my ovary questions here. Active one and four and five and negative three. So this is our change of coordinates matrix. So the next part I need to find ex breakfast, see two giving question. There's X equals five, one less three. You too. So the way to do that is to notice, except C is Come, I don't need to see times. The vector here is a five three. So this will be our previous meetings. Negative one. Why? For negative Three times I three, which iss 10. He left

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