linear algebra

Kat-M · #1 $Old$ January 11th, 2009, 02:52 PM

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 have no idea what to do. please help.

NonCommAlg · #2 $Old$ January 11th, 2009, 03:09 PM

Quote:

Originally Posted by Kat-M $View Post$

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 have no idea what to do. please help.

i think you mean $V=s(1) \oplus s(2) \oplus \cdots \oplus s(k),$ right?

Kat-M · #3 $Old$ January 11th, 2009, 03:49 PM

Quote:

Originally Posted by NonCommAlg $View Post$

i think you mean $V=s(1) \oplus s(2) \oplus \cdots \oplus s(k),$ right?

yes sorry i didnt know how to type $\oplus \$

NonCommAlg · #4 $Old$ January 11th, 2009, 04:27 PM

it's really easy: for any $1 \leq j \leq k$ let $T_j=T|_{s(j)}$ and $\sigma_j$ be a basis for $s(j)$ such that $[T_j]_{\sigma_j}=B_j.$ then $\sigma=\bigcup_{j=1}^k \sigma_j$ is a basis for $V$ and $[T]_{\sigma}$ is the block matrix given in your problem.

note that we keep the order of elements in $\sigma$ as they are in each $\sigma_j.$