Finding matrix representation of a linear map

Question

Let $ T : P_2(\mathbb{R}) \to P_2(\mathbb{R})$ be the linear map defined by $$ T(ax^2 +bx+c)=(a+b)x^2 +(b+c)x+(a+c). $$ Find the matrix for $T$ relative to the basis $B = \{x^2, x, 1\}$ (using the same basis $B$ for both the domain and codomain (or “target space”) of $T$).

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When I try this problem, I don't understand how to find matrix $T$ when there are variables mixed in with $x$.

Answer 1

Polynomials of degree 2 are an abstract vector space that is isomorphic to the vector space $\mathbb R^3$

The column vector $\begin{bmatrix} a\\b\\c \end{bmatrix}$ is the representation of the polynoimal $ax^2 + bx+ c$ with respect to the basis $\{x^2, x, 1\}$

That is all of the $x^2$ terms will be in the first entry, all of the $x$ terms will be in the second entry, and all of the constant terms go in the 3rd entry.

If this is strange to you, prove that polynomials meet all of the axioms of vector spaces.

$T$ is a transformation that takes $(a,b,c)$ to $(a+b, b+c, a+c)$

There exists some matrix $T$ such that $T\begin{bmatrix} a\\b\\c \end{bmatrix} = \begin{matrix} (a+b)\\(b+c)\\a+c \end{matrix}$

What the matrix $T$?

$T = \begin{bmatrix} 1&1&0\\0&1&1\\1&0&1\end{bmatrix}$

Answer 2

HINT

In vector form in the basis $B = \{x^2, x, 1\}$ we have that

$$T(a,b,c)\to(a+b,b+c,a+c)$$

thus

$$M_T=\begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{bmatrix}$$