Hey so I'm having trouble with this problem even though it should be pretty simple ...



Just going to use the notation for the rows and columns of matrices as [ a11 a12 a21 a22 ]

I think I have to do something where I have scalar values k, such that

k1*[ a 0 0 0 ] + k2*[ 0 b b 0 ] +k3*[ 0 0 0 c ] = 0 ?

And then I will prove that this spans all matrices [ a b b c ] ? Or something like that I'm not sure really I'm kind of lost

it's been a while, but i think you just have to show they span every matrix in your domain.

so:

k1*[ 1 0, 0 0 ] + k2*[ 0 1, 1 0 ] +k3*[ 0 0, 0 1 ] = [a b, b c]

then a = k1, b = k2, c = k3

could be wrong though.

maybe you have to show they're linearly independent? not sure

This post was edited by carteblanche on Sep 17 2014 06:40pm

So a = k1, b = k2, c = k3 shows that the given matrices span every matrix in the vectorspace?

you must prove that every single 2x2 symetric matrix can be written unambiguously as a linear combination of those 3 matrix
what carteblanche wrote is correct

Is he asked to prove that those 3 matrices form a base of the subspace of symmetric matrices, or only that they generate it ?

the span is ALL the linear combination of your matrices. to show the linear combinations you have

S =a1[1 0;0 0] + a2 [0 0; 0 1]+a3 [0 1;1 0] where S is the span

S = [a1 a3; a3 a2]
the matrix above shows all the linear combinations. now to show that it spans all the SYMMETRIC 2x2 matrices you have to show the transpose of S is S
To compute the transpose you just have to swap the rows into columns.

Edit: you should end up with [a1 a3;a3 a2]^T = [a1 a3;a3 a2]

This post was edited by cdexswzaq on Sep 20 2014 08:16pm