Block Diagonal Basis for Direct Sum of Linear Transformations

Theorem

• Suppose $T$ can be written as a Direct Sum of Linear Transformations $T_1\oplus \cdots \oplus T_k$
• Then, there exists a basis $\beta$ such that $[T{]}_{\beta }$ is Block Diagonal

Proof

1. Suppose $T$ can be written as a direct sum $T_1\oplus \cdots \oplus T_k$
2. Then, $V=W_1\oplus \cdots \oplus W_k$ such that $T_i$ is Invariant on $W_i$
3. let ${\beta }_i$ be a basis for $W_i$ and $\beta ={\bigcup }_{i=1}^k{\beta }_1$