Python Lab 8: Linear maps and change of basis

Open 08_Change_of_Basis.ipynb and enter s=$s, k=$k, p=$p, and q=$q. Let B=\left\{ \begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}$s\\1\end{bmatrix} \right\},\qquad [T]_B=\begin{bmatrix}2 & 0\\0 & $k\end{bmatrix},\qquad [x]_B=\begin{bmatrix}$p\\$q\end{bmatrix}.

1. Convert [x]_B to standard coordinates. The first component is first standard coordinate of x and the second component is second standard coordinate of x .

   $xStd1=$xStandardOne and $xStd2=$q

2. The change-of-basis matrix is P=\begin{bmatrix}1 & $s\\0 & 1\end{bmatrix}. Which formula gives the standard matrix M of T?

   formula for the standard matrix after change of basis M=P^{-1}[T]_BP M=P[T]_BP^{-1} M=P+[T]_B M=P^T[T]_BP $similarityChoice.selectedIndex=2

3. Your calculation should give M=\begin{bmatrix}2 & m_{12}\\0 & $k\end{bmatrix}. Enter m_{12}= upper right entry of the standard matrix .

   $m12Input=$mTwelve

4. Compute T(x) in standard coordinates. The first component is first standard coordinate of T of x and the second component is second standard coordinate of T of x .

   $tx1Input=$txOne and $tx2Input=$txTwo

5. Why do M and [T]_B have the same eigenvalues?

   reason two representations have the same eigenvalues They are equal entry by entry. Every two-by-two matrix has eigenvalues 2 and $k. They are similar matrices representing the same linear map. Both matrices are orthogonal. $eigenChoice.selectedIndex=3