This is a linear algebra textbook question:
If $W$ is a $T$-invariant subspace, we denote by $T: W \rightarrow W$ be the map defined by $T_W (v) = T(v)$.
(a) Let $V$ be finite-dimensional and $W$ a $T$-invariant subspace of $V$. Show that there exists a polynomial $g(t)$ such that $$\det(T − tI_V ) = g(t) \det(T_W − tI_W ).$$
To begin this, I'm thinking these determinants would yield the characteristic polynomial of $T$ in subspace $V$ and $T$ in subspace $W$. Then based on that polynomial we should be able to determine $g(t)$. However, I'm not sure at all what $\det(T − tI_V )$ or $\det(T_W − tI_W )$ should look like. Could someone please assist in continuing this thought process and explain what they should look like to continue this proof?