# Constructing a linear map given by the matrix

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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You wrote: This is referring to the set of continuous functions with degree less than or equal to 2. Did you mean polynomials of degree $\le2$? – Martin Sleziak Sep 26 '12 at 11:52
He must have, @MartinSleziak , otherwise he'll have to explain what does "degree of continuous function" mean in this context. – DonAntonio Sep 26 '12 at 12:08
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). – Marc van Leeuwen Sep 26 '12 at 12:45

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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You did not use the basis $1,x,x^2$ that was mentioned in the question, so the answer will need modification. – Marc van Leeuwen Sep 26 '12 at 12:42
Of course I did but since it wasn't given in any specific order, as basis usually are, I chose to use $\,\{x^2,x,1\}\,$ . This, of course, is completely unimportant: one can take the vector representing a polynomial "upwards" or "downwards". It's just a matter of being consistent. – DonAntonio Sep 26 '12 at 14:51
There's no point that I should be making a big fuss about. But the question does mention $(1~~x~~x^2)$ as a "coordinate system", in that order; and it affects the interpretation of the matrix. The formulation of the question is not very fortunate though, as my comment there indicates. – Marc van Leeuwen Sep 26 '12 at 16:02
Thanks for your help and clarification, yes I agree that the question was poorly worded but I took it from there! – Peter Sep 27 '12 at 2:22