Thread: 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]

i have no idea what to do. please help.

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 have no idea what to do. please help.

i think you mean right?

Originally Posted by NonCommAlg

i think you mean right?

yes sorry i didnt know how to type

it's really easy: for any let and be a basis for such that then is a basis for and is the block matrix given in your problem.

note that we keep the order of elements in as they are in each