[SOLVED] Bonus Question Linear Algebra

Hello people.

I just had a midterm today and did really well. There was a bonus question on the test that I had no idea how to do. Could someone please show me?

Prove or give a counter example:

a) If S and T are subspaces of R^n, then S (union) T is also a subspace R^n

is this b/c every element of s and every element of T are elements of R^n, so every element of S U T is also an element of R^n?

b) If S and T are subspaces of R^n, then S (intersection) T is also a subspace of R^n

is this b/c every element of s and every element of T are elements of R^n, so every element of S (intersection) T is also an element of R^n?

lol, I guess this is a bonus bonus question, but my prof made this a try it yourself question in his notes and i have no clue how to do it....

using the fact that rank(AB) <= rank(B) prove that if A and B are square matrices satisfying AB = I(sub)n then A^-1 = B and B^-1 = A

all i know so far is that rank is the number of pivot columns of a matrix

idk why our prof is introducing if then statements now, it's the last chapter in our textbook!

anyway, help would be much appreciated!