# Characteristic Polynomial of a Linear Map

I am hoping for some help with this question from a practice exam I am doing before a linear algebra final.

Let $T_1, T_2$ be the linear maps from $C^{\infty}(\mathbb{R})$ to $C^{\infty}(\mathbb{R})$ given by

$$T_1(f)=f'' - 3f' + 2f$$ $$T_2(f)=f''-f'-2f$$

(a) Write out the characteristic polynomials for $T_1$ and $T_2$.

(b) Compute the composition

(c) Find a vector in $\ker(T)$ which is not a linear combination of vectors in $\ker(T_1)$ and $\ker(T_2)$.

I know that the characteristic polynomial of an $n \times n$ matrix is the expansion of $$\det(A - I \lambda ).$$

Where I'm stuck here is finding the matrices for $T_1$ and $T_2$. I know it's silly, but I am used to applying these ideas to transformations of the form $T_A: \mathbb{R}^n \to \mathbb{R}^m$. Where $A$ is an $m \times n$ matrix and $T_A(\mathbf{x})= A \mathbb{x}$. Then $A$ is given by

$$A=[T(\mathbf{e}_1) \vdots T(\mathbf{e}_2) \vdots \cdots \vdots T(\mathbf{e}_n)].$$

Then I would find the characteristic polynomial of $A$. Now, I know that these ideas apply to abstract vector spaces like $C^{\infty}(\mathbb{R})$, but for some reason I cannot bridge the gap intuitively in the case of transformations. My textbook covers finding the matrix of a transformation for vectors in $\mathbb{R}^n$ but not for abstract vector spaces.

I am having the same problem with another question from the same practice exam:

Let $P_2$ be the vector space of polynomials of degree $2 \leq 2$, and define a linear transformation $T: P_2 \to P_2$ by $T(f)=(x+3)f' + 2f$. Write the matrix for $T$ with respect to the basis $\beta = (1,x,x^2)$.

Once again I do not know how to write the matrix for $T$. I would really appreciate any help understanding these concepts. Thanks very much.

First, for every $v\in\beta$, write $T(v)$ in the basis $\beta$: \begin{align} T(1) &= 2\cdot 1+0\cdot x+0\cdot x^2 \\ T(x) &= 3\cdot 1+3\cdot x+0\cdot x^2 \\ T(x^2) &= 0\cdot 1+6\cdot x+3\cdot x^2 \end{align} Now, the scalars appearing in these equations become the columns of $[T]_\beta$: $$[T]_\beta= \begin{bmatrix} 2 & 3 & 0 \\ 0 & 3 & 6 \\ 0 & 0 & 3 \end{bmatrix}$$ Of course, this matrix encodes a lot of information about the linear transformation $T$. For example, we can read off the eigenvalues as $2$ and $3$.