Question

I have a vector space V of all symmetric 2x2 matrices. The bases are S=([1 0; 0 0],[0 1;1 0],[0 0; 0 1]) and B = ([-1 2; 2 -2],[1 -1; -1 1],[2 -1; -1 2]) I need to find a transition matrix Ps,b
I don't want someone doing it for me but I need some guidelines on how to go about doing this. I find it a bit difficult since there's no standard basis.

Answer 1

S is the de facto standard basis here. But in any case, take any member of the basis B, and try and write it as a linear combination of the vectors (i.e., matrices) in S.

I think you'll find it surprisingly straightforward to do that.

Then the transition matrix will just be the matrix of coefficients (or its inverse to go the other way).

Answer 2

For example, call the members of S in the order you have written them s1, s2, s3; and the members of B, b1, b2, b3.

Then b1 = -s1 + ....

Answer 3

Thanks james, I'd thought S was the standard matrix but the vector [ 0 1; 1 0] threw me off, I though they all had to be along the lines of [1 0 0 0] [0 1 0 0] etc.

Answer 4

I'm a bit confused on how to write a member of my bases B as a linear combination of the 3 vectors in S. I've done examples with S having two vectors and B having two vectors, which was straight forward. Any advice on how to do this?

Answer 5

I'll give you the first one.
b1 = -s1 + 2s2 - 2s3

See that?

Answer 6

Oh alright, yeah I'd actually written that but didn't think it was what I should've been doing. So do I do the same for all the matrices in B?

Answer 7

Yes of course

Answer 8

And think of them not as matrices but as vectors

Answer 9

I'm sorry if these questions seem a bit daft. I find linear algebra really abstract and just difficult to understand. Thanks again :)

Answer 10

np