2
$\begingroup$

Linear transformation $T_1:M_2(\mathbb R)\to\mathbb R^2$ is defined such that $ T_1(\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}) = \begin{pmatrix} a+d \\ b+c \\ \end{pmatrix} $

  1. If $\mathcal E = (\begin{pmatrix}1 & 0 \\0 & 0 \\ \end{pmatrix},\begin{pmatrix}0 & 1 \\0 & 0 \\ \end{pmatrix},\begin{pmatrix}0 & 0 \\1 & 0 \\ \end{pmatrix},\begin{pmatrix}0 & 0 \\0 & 1 \\ \end{pmatrix})$ is the basis of $M_2(\mathbb R)$ and $\mathcal F = (\begin{pmatrix}1\\0\\\end{pmatrix},\begin{pmatrix}0\\1\\\end{pmatrix})$ is the basis of $\mathbb R^2$ , find the transformation matrix for $T_1$

  2. If $\mathcal E' = (\begin{pmatrix}1 & 0 \\0 & 1 \\ \end{pmatrix},\begin{pmatrix}0 & 1 \\-1 & 0 \\ \end{pmatrix},\begin{pmatrix}1 & 0 \\1 & 0 \\ \end{pmatrix},\begin{pmatrix}0 & 0 \\1 & 1 \\ \end{pmatrix})$ is another basis of $M_2(\mathbb R)$ , find the matrix change of basis $T_2:\mathcal E\to\mathcal E'$

  3. If $\mathcal F' = (\begin{pmatrix}1\\1\\\end{pmatrix},\begin{pmatrix}-1\\1\\\end{pmatrix})$ is another basis of $\mathbb R^2$ , find the matrix change of basis $T_3:\mathcal F'\to\mathcal F$

  4. With $\mathcal E'$ and $\mathcal F'$ as the basis for $M_2(\mathbb R)$ and $\mathbb R^2$ , find the transformation matrix $T_4:M_2(\mathbb R)\to\mathbb R^2$

I've found :

  • $T_1=\begin{pmatrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ \end{pmatrix}$ by applying transformation $T_1$ to each of $M_2(\mathbb R)$ basis for question 1

  • $T_3=\begin{pmatrix}1 & -1 \\ 1 & 1 \\ \end{pmatrix}^{-1}=\begin{pmatrix}1/2 & 1/2 \\ -1/2 & 1/2 \\ \end{pmatrix}$ for question 3

If $T_1=\begin{pmatrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ \end{pmatrix}$ is true how do I transform another matrix from $M_2(\mathbb R)$?

Because it is going to be $ \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ \end{pmatrix} \times \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = \begin{pmatrix} a+d \\ b+c \\ \end{pmatrix} $

but it's wrong according to the definition of matrix multiplication.

So I'm also not sure whether $T_2=\begin{pmatrix}1&0&0&1&1&0&0&0 \\ 0&1&-1&0&1&0&1&1 \\ \end{pmatrix}$.

And for question 4 I understand that $\require{AMScd}$ \begin{CD} \mathcal E @> T_1 >> \mathcal F\\ @V T_2 V V\circlearrowleft @AA T_3 A\\ \mathcal E' @>> T_4 > \mathcal F' \end{CD} so is the relation $T_1=T_3 T_4 T_2 \Longleftrightarrow T_3^{-1} T_1 T_2^{-1} = T_4$ correct?

0 Answers 0