Question

Consider the vector space p2 of polynomials of degree at most 2. let t be the linear transformation given by t(p(x)) = p(x − 1) + p(x + 1). find the matrix of t with regard to the basis {1, x, x 2 }.

Answer 1

Observe what happens to the basis components under the transformation $T$:

$T(1)=1+1=2=2\cdot1+0\cdot x+0\cdot x^2$
$T(x)=(x-1)+(x+1)=2x=0\cdot1+2\cdot x+0\cdot x^2$
$T(x^2)=(x-1)^2+(x+1)^2=2x^2+2=2\cdot1+0\cdot x+2\cdot x^2$

So we can represent $T$ with the matrix

$\mathbf T=\begin{bmatrix}2&0&2\\0&2&0\\0&0&2\end{bmatrix}$

To check this answer: If we take an arbitrary polynomial $\mathbf p\in\mathbb P_2$, i.e. the vector

$\mathbf p=\begin{bmatrix}c\\b\\a\end{bmatrix}$

corresponding to the polynomial $p(x)=c+bx+ax^2=c\cdot1+b\cdot x+a\cdot x^2$, we get

$\mathbf{Tp}=\begin{bmatrix}2a+2c\\2b\\2a\end{bmatrix}=(2a+2c)\cdot1+2b\cdot x+2a\cdot x^2$

Substituting $\mathbf p$ into the rule above, we get

$p(x-1)+p(x+1)=a(x-1)^2+b(x-1)+c+a(x+1)^2+b(x+1)+c$
$p(x-1)+p(x+1)=(2a+2c)+2bx+2ax^2$

as expected.