Question

If we know S and T are subspaces of a vector space V, how do we show S+T is a subspace of V?

Answer 1

For this, you need to show three things.
1. \(0\in S+T\).
2. If \(M,M'\in S\), \(N,N'\in T\), then \((M+N)+(M'+N')\in S+T\)
3. If \(a\) is a scalar, \(M\in S\), \(N\in T\), then \(a(M+N)\in S+T\).

Answer 2

What are M' and N'?

Answer 3

\(M'\) and \(N'\) are just more elements of the vector space. I just ran out of letters I like to use. I suppose I could have used lower case letters as well.

I think you can manage the first condition on your own. As for the second, note that \[(M+N)+(M'+N')=(M+M')+(N+N').\]Since \(S,T\) are subspaces, \(M+M'\in S\) and \(N+N'\in T\). So \((M+N)+(M'+N')\in S+T\).

Answer 4

Okay. I will give it a go and see what I get...