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**Edit complete, but it doesn't seem as though I can change the title.**

The latex arrows next to the 'P' aren't showing up for me but they're supposed to be left arrows

The latex arrows next to the 'P' aren't showing up for me but they're supposed to be left arrows

## Homework Statement

Let B and C be bases of R^2. Find the change of basis matrices [itex]P_{B \leftarrow C}[/itex] and [itex]P_{C\leftarrow B}[/itex]

[itex]B={\begin{pmatrix}3\\1\end{pmatrix}, \begin{pmatrix}2\\2\end{pmatrix}}, C={\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}5\\4\end{pmatrix}}[/itex]

To find the change of basis matrix from C to B [itex]P_{B \leftarrow C}[/itex] , I followed the steps in Lay's Linear Algebra book and found the coordinates of the B vectors relative to C.

So I solved this system:

[itex]\begin{pmatrix}3\\1\end{pmatrix}=r_1\begin{pmatrix}1\\0\end{pmatrix}+r_2\begin{pmatrix}5\\4\end{pmatrix}[/itex]

[itex]\begin{pmatrix}2\\2\end{pmatrix}=s_1\begin{pmatrix}1\\0\end{pmatrix}+s_2\begin{pmatrix}5\\4\end{pmatrix}[/itex]

where r, s are real numbers.

Doing so, I got the 2x2 matrix

[7/4 -2/4]

[1/4 2/4]

(can't get matrices to work in latex)

But in the solutions, this is the change of basis matrix for going from C to B, i.e. [itex]P_{C\leftarrow B}[/itex]. Have I misinterpreted something? The steps I'm referring to are based on this theorem on page 273 of the 3rd edition of the text by Lay:

Let [itex]B=[b_1,..,b_n][/itex] and [itex]C=[c_1,..,c_n][/itex] be bases of a vector space V. Then there is a unique nxn matrix [itex]P_{B \leftarrow C}[/itex] such that [itex][x]_c=P_{B \leftarrow C}[x]_B[/itex]. The columns of [itex]P_{B \leftarrow C}[/itex] are the C-coordinate vectors of the vectors in the basis B. That is, [itex]P_{B \leftarrow C}=[[b_1]_c...[b_n]_c][/itex]

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