Finding the basis for a vector space

[aanandpatel]

Homework Statement

Find a basis for the following vector space:

The set of 2x2 matrices A such that CA=0 where C is the matrix : 1 2
3 6

The Attempt at a Solution

I multiplied C by a general 2x2 matrix : a b and got 4 equations but two of these equations are the
c dsame and it seems as I am going around in circles. I know I need to find a set that is linearly independent and spans the original set but I'm not sure how to proceed.

Help would be greatly appreciated :)

Cheers

 

[CompuChip]

That sounds exactly like what you want. Clearly not any 2 x 2 matrix will do, and indeed the two equations you have will fix two relations between a, b, c and d. But since there is not a single unique solution, you should also have at least one parameter which you can vary.

Talking with a practical example is probably clearer, so can you show us which equations you got?

Have a matrix by the way (quote my post to see the code): $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

 

[aanandpatel]

sorry had no idea how to matrices on the forum

C = $\begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}$

A(my general matrix) = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

CA=0 so when I multiplied, I got

a+2c=0
b+2d=0
3a+6c=0
3b+6d=0

But two of those equations are the same and I don't know what to do after that.

 

[CompuChip]

Right, so if you get all the information you can from those equations, you can write your matrix A in terms of two variables only - for example, just a and b.

What does A look like then?

 

[aanandpatel]

Makes sense now - got it!

Thanks a lot - help was much appreciated :)