Calculate the minimal and characteristic polynomial of $T$ given the $F[x]$-structure

Question

The question that I have (modified from the original problem) is the following:

Say $ V =\mathbb{C}[x]/(x-2) \oplus \mathbb{C}[x]/(x^2)$. What is the minimal and characteristic polynomial of the corresponding linear transformation $T: V \to V$, where $ T $ is multiplication by $ x $?

I have problem converting the $ \mathbb{C}[x] $-module structure of $V$ back to $T$. In particular, I have trouble understanding or seeing what $ T $ is like, or how does $ V$ look as a $ \mathbb{C} $-vector space. It is very confusing to me.

Any help is appreciated.

Answer 1

Note that $V$ is a vector space consisting of polynomials of the form $(p(x),q(x))$ modulo the appropriate relations. So really, the elements look like $(p(x) + (x-2),q(x) + (x^2))$ (I leave off the ideals for convenience). $V$ has the basis $$ \mathcal B = \{(1,0),(0,x),(0,1)\} =: \{v_1,v_2,v_3\} $$ To see that this is really a basis, note that each element of $V$ can be uniquely expressed as $$ (p(x),q(x)) = (a_0, b_1x + b_0) = a_0(1,0) + b_1 (0,x) + b_0(0,1) $$ We find that $$ T(v_1) = x(1,0) = (x,0) = (2,0) = 2v_1\\ T(v_2) = x(0,x) = (0,x^2) = (0,0) = 0\\ T(v_3) = x(0,1) = (0,x) = v_2\\ $$ With that, we find that the matrix representation of $V$ with respect to this basis is $$ \pmatrix{2&0&0\\0&0&1\\0&0&0} $$ Hopefully that helps.

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The minimal and characteristic polynomial will both be $x^2(x-2)$.

Answer 2

By the CRT, $\mathbb{C}[x] / (x - 2) \oplus \mathbb{C}[x] / (x^2) \cong \mathbb{C}[x] / (x^2(x - 2))$. The min and char polynomials are therefore $x^2(x - 2)$.

Answer 3

For every summand $\def\C{\Bbb C}\C[x]/P$ with $P\in\C[x]$ monic, the matrix on the basis of (images of) monomials $x^i$ for $0\leq i<\deg P$ of multiplication by (the image of) $x$ is the companion matrix for$~P$. It has minimal and characteristic polynomials both equal to$~P$. All you need to do it take the product of these factors$~P$ for the characteristic polynomial, and their least common multiple for the minimal polynomial. In the example the two polynomials are relatively prime, so it gives the same result in both cases.