<s>[INST] Linear Transformation Matrix with polynomials A linear transformation $T : P_2 \to P_2$ has matrix with respect to $S$ given by:
$$[T]\,( S) = \begin{bmatrix}
1/2&-3&1/2\\
-1&4&-1\\
1/2&2&1/2\\
\end{bmatrix}
$$
How do you find $T(a+bx+cx^2)$?
Thank you!!
 [/INST] Every polynomial $S$ of degree $2$ s.t. $S \in P_2$  can be represented as a vector in three dimensional space: 
$$
S = a + bx + cx^2 \quad  \iff \quad 
\begin{bmatrix}
a \\ b \\ c \\
\end{bmatrix}
\cdot
\begin{bmatrix}
1 \\ x \\ x^2 \\
\end{bmatrix}, 
$$
therefore we can associate $S$ with a 3D vector
$$
S  \longleftrightarrow 
\begin{bmatrix}
a \\ b \\ c \\
\end{bmatrix}.
$$
When you have the polynomial represented as a vector, applying linear transformation $T$, given in form of a matrix, is a piece of cake:
$$
[T]\,( S) = \begin{bmatrix}
1/2&-3&1/2\\
-1&4&-1\\
1/2&2&1/2\\
\end{bmatrix}
\begin{bmatrix}
a \\ b \\ c \\
\end{bmatrix}
=
\begin{bmatrix}
\frac{1}{2}a - 3 b + \frac{1}{2}c \\
 -a + 4 b -c \\ 
\frac{1}{2}a +2 b + \frac{1}{2}c \\
\end{bmatrix}
,
$$
which will correspond to the polynomial
$$
T\left( a+bx+cx^2\right)  = 
\left(\frac{1}{2}a - 3 b + \frac{1}{2}c\right)  +
\left( -a + 4 b -c \right) x + 
\left(\frac{1}{2}a +2 b + \frac{1}{2}c \right)x^2
$$
</s>