What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial? I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the characteristic polynomial.)
I guess just like how we defined $\chi_T=\chi_A$, I can think of it as evaluating at $A= {}_B[T]_B$ with respect to some basis $B$. However, wouldn't $\chi_T(T)$ be different depending on what basis you choose in this case?
Thanks.
 A: It's pretty straightforward. If $V$ is a $k$-vector space, and $T\in\mathrm{End}(V)$ is some $k$-linear map $T:V\to V$, then the polynomial
$$p=a_0+a_1x+\cdots+a_nx^n\in k[x]$$
evaluated at $T$ is just
$$a_0+a_1T+\cdots + a_nT^n\in\mathrm{End}(V)$$
where $T^k$ is $T$ composed with itself $k$ times, and $aT$ is the element of $\mathrm{End}(V)$ defined by
$$(aT)(v)=a\cdot T(V)$$

Here is a more abstract take on things. It is easy to check that the collection of all linear maps $T:V\to V$, denoted by $\mathrm{End}(V)$, is a (non-commutative) ring, where addition is pointwise addition:
$$(T_1+T_2)(v):=T_1(v)+T_2(v)$$
and multiplication is composition:
$$(T_1\cdot T_2)(v):=T_1(T_2(v))$$
Now choose some particular $T\in\mathrm{End}(V)$. By the universal property of the polynomial ring $k[x]$, we can define a ring homomorphism $k[x]\to\mathrm{End}(V)$ by simply declaring that $x$ should go to $T$; everything else about how the homomorphism acts is then determined. The result is the evaluation homomorphism
$$\mathrm{ev}_T:k[x]\to\mathrm{End}(V),\qquad \mathrm{ev}_T(p)=p(T)$$
A: Polynomial rings are characterized by a universal property which ensures that this makes sense.
Let $k$ be a field and let $V$ be a $k$-vector space. Then the collection of endomorphisms of $V$ ($k$-linear maps from $V$ to itself) forms a ring, $\text{End}_k(V)$, with pointwise addition and composition as multiplication. Given any transformation $T\in \text{End}_k(V)$, there is a unique ring homomorphism $\epsilon_T:k[x]\rightarrow \text{End}_k(V)$ sending constant polynomials to corresponding scalar transformations and sending the indeterminate $x$ to $T$. This is called the evaluation map for $T$.
The "evaluation" of a polynomial $p\in k[x]$ at $T\in \text{End}_k(V)$ is by definition $\epsilon_T(p)$, and it is often denoted by "$p(T)$".
A: The characteristic polynomial of the matrix representing a transformation $T$ does not depend on the choice of basis.
In particular: for any basis $C$ and a fixed basis $B$, we have $[T]_C = S[T]_BS^{-1}$ for some invertible matrix $S$.  We then have
$$
\det([T]_C - tI) = 
\det(S[T]_BS^{-1} - tI) = 
\det(S([T]_B-tI)S^{-1}) = \\
\det(S)\det([T]_B-tI)\det(S^{-1}) = 
\det([T]_B - tI)
$$
