Basis of M2,2 is not spanning set of trace zero matrices?

Question

given set of matrices:

S={[1 0; 0 0];[0 1; 0 0];[0 0; 1 0];[0 0; 0 1]}

I have to explain why S is not a spanning set of matrices with trace zero, matrices of:

V be the subspace of M2,2: V = {[a b; c d] |a+d=0,a,b,c,d∈R}

however according to my assumptions: a[1 0; 0 0]+b[0 1; 0 0]+c[0 0; 1 0]-a[0 0; 0 1] ∈ V.

I'm getting the combination of S spanning V. Are my calculations wrong or something, please advice.

Answer 1

$S$ is said to be a spanning set of $V$ if its linear span is exactly equal to $V$. Your $S$ spans the whole space $M_{2,2}$, which is strictly larger than $V$. So, $S$ not a spanning set of $V$.