subspaces: Is S-intersect-T a subspace of V?

[Clifford]

S and T are subspaces of V

is S intersection T a subspace of V?
is S union T a subspace of V?
is A = {w = u + 3v, uES, vET} a subspace of V?

Here is my solution, is this right or am I lacking information:

Let u and v be two vectors
uES, vES
uET, vET
u+v E S since S is a subspace
u+v E T since T is a subspace
thus u+v E S intersection T,
hence S intersection T is a subspace of V

I used the exact same proof for S union T, although replace the intersections with unions.

For the last question, A = {w = u + 3v, uES, vET} is a subspace of V since uES and vEt, and S union T is a subspace of V.

 

[pka]

Re: subspaces

Given a subset of a vector space, in order to show that set is a subspace you must show that it is closed with respect to vector addition and scalar multiplication.

Thus in #1 start with two vectors in \(\displaystyle S \cap T\) and show their sum is in \(\displaystyle S \cap T\).
Is that what you did? It does not appear to be.

#1 works because intersection is an “and”. But #2 fails because union is an “or”.
Given two vectors in \(\displaystyle S \cup T\) one might come from \(\displaystyle S\backslash T\) and the other from \(\displaystyle T\backslash S\) thus the sum may not belong to \(\displaystyle S\cup T\).
You need to find a counterexample.

In #3 A is a subspace.