Showing H is a Subspace of M2x2

[judahs_lion]

H = ([a,b;c,d] : a+d =0}

Dim(M_2x2)= 4, so a basis would have 4 components?

I got this far and am stuck.

[a, b ; c , -a] = a[1,0; 0,-1] + b[0, 1;0,0] +c[0,0; 1,0]

 

[Mark44]

H = ([a,b;c,d] : a+d =0}

Dim(M_2x2)= 4, so a basis would have 4 components?

A basis for M_2x2 would have 4 vectors/matrices, but how many would be in a basis for H?

I got this far and am stuck.

[a, b ; c , -a] = a[1,0; 0,-1] + b[0, 1;0,0] +c[0,0; 1,0]

Now, do these matrices span H? I.e., can every matrix in H be written as a linear combination of the three matrices above?

What's left to do is to show that H is a subspace of M_2x2. To do this, you need to show three things:
That the zero matrix is in H.
That if A and B are in H, then A + B is in H.
That if A is in H and c is a scalar, then cA is in H.

 

[judahs_lion]

Thank you