Let B = {b1, b2} and C = (c1, c2) be bases for a vector  V, and suppose b1 = -c1 + 4c2 and b2 = 5c1 – 3c2. Find the change of coordinate matrix for a vector space and find [x]c for x = 5b1 + 3b2.

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Let’s find change of coordinate matrix from B to C.

\(P_{C ← B} = \begin{bmatrix}[b_1]_c & [b_2]_c\end{bmatrix}\)

Here, B = {b1, b2} and c={c2, c2} are bases for V.

Given,

b1 = -c1 + 4c2

b1 = \(\begin{bmatrix}c_1 & c_2\end{bmatrix}\begin{bmatrix}-1\\ 4\end{bmatrix}\)

∴ [b1]c = \(\begin{bmatrix}-1\\ 4\end{bmatrix}\)

Also,

b2 = 5c1 – 3c2

b2 = \(\begin{bmatrix}c_1 & c_2\end{bmatrix}\begin{bmatrix}5\\ -3\end{bmatrix}\)

∴ [b2]c = \(\begin{bmatrix}5\\ -3\end{bmatrix}\)

∴ change of cordinate matrix from B to C is

\(P_{C ← B} = \begin{bmatrix}[b_1]_c & [b_2]_c\end{bmatrix}\)

= \(\begin{bmatrix}-1 & 5\\ 4 & -3\end{bmatrix}\)

Also, Given

x = 5b1 + 3b2

x = \(\begin{bmatrix}b_1 & b_2\end{bmatrix}\begin{bmatrix}5\\ 3\end{bmatrix}\)

∴ [x]B = \(\begin{bmatrix}5\\ 3\end{bmatrix}\)

Now,

[x]C = \(P_{C ← B} [x]_B\)

= \(\begin{bmatrix}-1 & 5\\ 4 & -3\end{bmatrix}\begin{bmatrix}5\\ 3\end{bmatrix}\)

=\(\begin{bmatrix}-5 + 15\\ 20-9\end{bmatrix}\)

=\(\begin{bmatrix}10\\ 11\end{bmatrix}\)

[x]C = \(\begin{bmatrix}10\\ 11\end{bmatrix}\)

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