the M,(R) is Mn(R)
#hello, can someone help me with this algebra problem?
ripe loom
wintry valve
Well, antisymmetric matrices have the following condition: a(i, j) = -a(j, i).
This imposes a condition: a(i, i) = 0.
As for the basis, we can just take n(n - 1)/2 matrices whose entries are just a pair of elements equal distance away from the diagonal equal to -1 and 1. So, the dimension of this subspace is n(n - 1)/2, less than half the dimension of the whole space (n^2).
Also, it's not that hard to prove that this is a subspace, so try that too, just for the exercise.