Answer to Question #7824 in Abstract Algebra for Vais

Question

Linear mathematics Question

The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.
The set of antisymmetric matrices A = { 0 a }
−a 0
where a belongs to R is a subset of M2,2.
Prove that A is a subspace of M2,2.

Answer 1

We have to show that if matrices
P = [ [0,p]; [-p,0] ]
and
Q = [
[0,q]; [-q,0] ]
belong to A, and t is a real number, then

P+Q, and tP
belong to A as well.

Notice that
P+Q = [ [0,p+q]; [-p-q,0] ] = [
[0,p+q]; [-(p+q),0] ]
and
tP = [ [0,tp]; [-tp,0] ]

Both matrices
are antisymmetric, so they belong to A,
and therefore A is a subspace of M2,2