Characteristic polynomial/Direct sum decomposition/Fact/Proof

{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Let ${\displaystyle {}u_{1},\ldots ,u_{k}}$ be a basis of ${\displaystyle {}U}$ and ${\displaystyle {}w_{1},\ldots ,w_{m}}$ be a basis of ${\displaystyle {}W}$; together they form a basis of ${\displaystyle {}V}$. With respect to this basis, ${\displaystyle {}\varphi }$ is described by the block matrix ${\displaystyle {}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}}$, where ${\displaystyle {}A}$ describes the restriction {{mathl|term= \varphi {{|}}_U |pm=}} and ${\displaystyle {}B}$ describes the restriction {{mathl|term= \varphi {{|}}_W |pm=.}} Then, using exercise, we get {{ Relationchain/display | \chi_{ M } || \det { \left( t \operatorname{Id} -M \right) } || \det { \left( t \operatorname{Id} -A \right) } \det { \left( t \operatorname{Id} -B \right) } || {{op:Characteristic polynomial|\varphi{{|}}_U |}} \cdot {{op:Characteristic polynomial|\varphi{{|}}_W |}} |pm=. }} |Closure= }} |Textform=Proof |Category=See }}