1. linear algebra

let T be a linear transformation from V to V. suppose V=s(1)+s(2)+,,,+s(k), where each subspace s(i) is invariant under T. if T can be represented on s(i) by the matrix Bi, show that T can be represented on V by the matrix

[B1 00000000]
[0 B2 000000]
[00 B3 00000]
[ ............... ]
[ 0000000 Bk]

2. Originally Posted by Kat-M
let T be a linear transformation from V to V. suppose V=s(1)+s(2)+,,,+s(k), where each subspace s(i) is invariant under T. if T can be represented on s(i) by the matrix Bi, show that T can be represented on V by the matrix

[B1 00000000]
[0 B2 000000]
[00 B3 00000]
[ ............... ]
[ 0000000 Bk]

i think you mean $\displaystyle V=s(1) \oplus s(2) \oplus \cdots \oplus s(k),$ right?
i think you mean $\displaystyle V=s(1) \oplus s(2) \oplus \cdots \oplus s(k),$ right?
yes sorry i didnt know how to type$\displaystyle \oplus \$
4. it's really easy: for any $\displaystyle 1 \leq j \leq k$ let $\displaystyle T_j=T|_{s(j)}$ and $\displaystyle \sigma_j$ be a basis for $\displaystyle s(j)$ such that $\displaystyle [T_j]_{\sigma_j}=B_j.$ then $\displaystyle \sigma=\bigcup_{j=1}^k \sigma_j$ is a basis for $\displaystyle V$ and $\displaystyle [T]_{\sigma}$ is the block matrix given in your problem.
note that we keep the order of elements in $\displaystyle \sigma$ as they are in each $\displaystyle \sigma_j.$