I've been learning linear algebra and I came across this question that I don't know how to solve. The question begins by letting $W_1$ and $W_2$ be subspaces of a finite-dimensional vector space $V$. Suppose that $V=W_1 \oplus W_2$. Then, for any $x∈V$ we may express $x$ uniquely as $x=w_1+w_2$ where $w_1∈W_1$ and $w_2∈W_2$. Let $R: W_1 \to W_1$ and $S: W_2\to W_2$ be linear transformations and define the linear transformation$T = \mathrm{det}\, R\, \mathrm{det}\, S$ given $T: V\to V$ by $T{\bf x}=R{\bf w}_1+S{\bf w}_2$
The question asks to prove $\mathrm{det}\, T = \mathrm{det}\, R\, \mathrm{det}\, S$
If anyone could give some guidance or help solve this problem, I would really appreciate it
Hint: If $A$ and $B$ are square matrices, then $$\det\pmatrix{A&0\\0&B}\ =\ \det A\cdot\det B\,.$$