I need help solving (b) in this math problem.
What is the relationship between the matrix f and the transformation matrix in the standard basis?
Thanks for any suggestions!
I need help solving (b) in this math problem.
What is the relationship between the matrix f and the transformation matrix in the standard basis?
Thanks for any suggestions!
Say $\gamma=[g_1,g_2]$, $\beta$ is the standard ordered basis, and $[T]_{\gamma}$ is the matrix from part $(a)$.
We can find the change-of-basis matrix from $\gamma$ to $\beta$ by writing $g_1$ and $g_2$ as linear combinations of ${1 \choose 0}, {0 \choose 1}$:
${5 \choose 4}=5{1 \choose 0}+ 4{0 \choose 1}$
${-4 \choose 5}=-4{1 \choose 0}+ 5{0 \choose 1}$
These coefficients form the columns of the change-of-basis-matrix from $\gamma$ to $\beta$.
$[Q]_{\gamma}^\beta= \begin{pmatrix} 5 & -4 \\ 4 & 5 \end{pmatrix} $
We can also find the change-of-basis-matrix from $\beta$ to $\gamma$ by reversing the process above (writing ${1 \choose 0}$ and ${0 \choose 1}$ as linear combinations of $g_1$ and $g_2$) or by taking the inverse of $[Q]_{\gamma}^\beta$.
In either case, $[Q]_{\beta}^\gamma=\begin{pmatrix} \frac{5}{41} & \frac{4}{41} \\ \frac{-4}{41} & \frac{5}{41} \end{pmatrix} $
Then we multiply the matrices in the proper order. (The matrix $[T]_{\beta}$ is the matrix representation of $f$ with respect to the standard ordered basis.)
$[T]_{\beta}=[Q]_{\gamma}^\beta[T]_{\gamma}[Q]_{\beta}^\gamma= \begin{pmatrix} 5 & -4 \\ 4 & 5 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} \frac{5}{41} & \frac{4}{41} \\ \frac{-4}{41} & \frac{5}{41} \end{pmatrix}= \begin{pmatrix} \frac{9}{41} & \frac{40}{41} \\ \frac{40}{41} & \frac{-9}{41} \end{pmatrix}$
I hope that helps!