Finding the Matrix of a Linear Transformation With Polynomial Basis

Question

If I have some given linear transformation $T:V\rightarrow V$ such that:

$T(p(x)) = p'(x) + x\,p(0) + (x+1)\,p(1)$,

where $V$ is the space of rational polynomials of degree at most 2 with basis $\{1 ,x,x^2\}$.

Now I'm trying to figure out what $T$ actually does. $p'(x)$ means derivative of $p(x)$ and $p(0)$ is some rational number when $p(x)$ is evaluated in zero,which is just the constant term of $p(x)$.

What I'm confused by is this $p(1)$. Do I need an extra parameter when applying $T$ to a basis vector ?

For example: $T(1)=0+x+(x+1)\,a$

where $a \in \mathbb{Q}$.

Answer 1

We'll consider how $T$ maps the given basis $\{1,x,x^2\}$ in order to find the matrix of this transformation in said basis:

$$T(1) = 0+1*x+(x+1)*1 = 2x+1$$ $$T(x) = 1 + 0*x + (x+1)*1 = x+2$$ $$T(x^2) = 2x + 0*x + (x+1)*1 =3x+1 $$

So, we have $$T = \begin{bmatrix} 1&2&1\\2&1&3\\0&0&0\end{bmatrix}$$

Answer 2

You have $p (x):=1$, the constant polynomial. Then $T (1)=0+x1+(x+1)1=2x+1$.