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## Homework Statement

i know how to find the basis of a subspace of R

^{2}or R

^{3}but I can't figure out how to find the basis of a subspace of something like R

^{2,2}.

I even have an example in my book which i managed to follow nearly till the end but not quite...

Given matrix:

A=

6 -9

4 -4

show that the subset F={X (of R

^{2,2}) | AX=XA } is a vector subspace of R

^{2,2}and find its basis

X =

x1 x1

x3 x4

## Homework Equations

## The Attempt at a Solution

I followed the example through by multiplying the matrices in the eq AX-XA=0 and found the solution for the associated system of equations which after reducing it is:

4*x1 - 12*x3 - 4*x4 = 0

9*x1 +12*x2 - 9*x4 = 0

so:

x1 = 3*x3 + x4

x2 = -9/4*x3

x3 = x3

x4 = x4

then the example just says:

therefore the dimension of subset F is 2 and the basis are the 2 matrices:

12 -9

4 0

and

0 1

1 0

I don't know where these numbers come from please help.

I looked through the book and it only shows how to find basis for R

^{n}but not R

^{n,n}