# Determine which of the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$.

I'm having a bit of trouble showing that the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$. I know that I need to show that they are closed under addition and multiplication, but I do I need to show it for the $nn$ and the $nm$ case? If so, how do I show it for the $nm$ case? Furthermore, is part d the set of $n$-tuples?

a) The symmetric matrices

b) The nonsingular matrices

c) The diagonal matrices

d) $\{x \mid \sum_{j=1}^{n}x_j =0\}$

• As for the first three (a,b,c), those matrices aren't even defined in the $nm$ case (if n is not equal to m). Part d is the set of n-tuples whose components sum to zero (at least i think..it doesn't really make sense written as it is) – Franco Apr 19 '14 at 0:38

• If I understand it correctly, $x_j$ are column vectors and there are n of them, also each of them is n-dimensional meaning $x_j=(x_{j,1},\dots x_{j,n})$. – L.G Apr 19 '14 at 0:50