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$W = \{A \in M_{2\times2} (\mathbb{R}): A_{11} = A_{12}\text{ and }A_{22} = A_{21}\}$ is isomorphic to $P_1 (\mathbb{R})$.

Show the transformation matrix in relation to the canonical basis of the respective spaces.

I'm in doubt about the "canonical basis" of $W$ and $P_1$. Well, I know that the caninocal basis of $P_1$ is $\{1, x\}$. Intuitively, I think the canonical basis of $W$ is the set of the matrices

\begin{bmatrix} 1 & 1\\ 0 & 0 \end{bmatrix}

and

\begin{bmatrix} 0 & 0\\ 1 & 1 \end{bmatrix}

I can understand the transformation and write it with $1 \times 2$ matrices (and the transformation as a $2 \times 2$ matrix), for example. but I don't know how to show the transformation matrix in the canonical basis of each set.

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  • $\begingroup$ You may want to notice my edits. There's no need to write $2x2$; I changed it to $2\times2$. And the curly braces in $\{A\in M_{2\times2} : \cdots\cdots\cdots\}$ were invisible for a reason, which I corrected. Also, your matrices are now bounded by square brackets. And where you had $A_{12} and A_{22}$, you now see $A_{12}\text{ and }A_{22}$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Apr 22 '14 at 20:47
  • $\begingroup$ You're welcome. I had realized some problems but I didn't know how to fix them. Thanks. $\endgroup$ – user35477 Apr 22 '14 at 21:23
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A canonical bases is something like a "natural/obvious choice" of a basis. So you cannot prove, that something is a canonical bases, you can just state, that most of the mathematicians would use take a given basis as a canonical one.

Your choice for the canonical bases are right. Note, that if you take a transformation $f$ which maps

$\begin{align}\left(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}\right) & \mapsto 1 \\ \left(\begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix}\right) & \mapsto x \end{align}$

then the matrix of $f$ is $\left(\begin{matrix} 1 & 0 \\ 0 & 1\end{matrix}\right)$.

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