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I have a question to do with linear algebra.

Consider the linear map $T:\mathbb{R}[x]\le _2 \to\mathbb{R}^2$, given by the matrix:

$ \begin{pmatrix} 1 & 1 & 2\\ 2 & 0 & 3 \end{pmatrix} $ (sorry for my terrible formatting but it's my first time posting. This is referring to the set of continuous functions with degree less than or equal to 2.)

Find the linear map with respect to the coordinate system $\begin{pmatrix}1 & x & x^2\\\end{pmatrix}$:$\mathbb{R}^3\to\mathbb{R}[x]\le _2$ (and the standard coordinate system $id:\mathbb{R}^2\to\mathbb{R}^2$).

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    The terminology in the question is confusing. I suppose the matrix is given with respect to the coordinate system (I would say basis) $(1~x~x^2)$ of $\mathbb{R}[x]\le _2$ (and the standard basis of $\mathbb{R}^2$), and the question is to describe the linear map directly (not using coordinates).2012-09-26

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This pressumes you identify any element (polynomial) in $\,\Bbb R[x]_2\,$ with a (column) vector in $\,\Bbb R^3\,$ :

$ax^2+bx+c\longrightarrow\begin{pmatrix}a\\b\\c\end{pmatrix}$

Then the linear map is given by

$T(ax^2+bx+c):=\begin{pmatrix}1&1&2\\2&0&3\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix}a+b+2c\\2a+3c\end{pmatrix}$

Can you take it fom here?

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    Thanks for your help and clarification, yes I agree that the question was poorly worded but I took it from there!2012-09-27