If S=sp(v1,v2) and T=sp(v3,v4) where v1=(1,1,1,0)^t, v2=(0,-1,1,0)^t, v3=(1,0,0,1)^t and v4=(0,2,0,-1)^t find all vectors which lie in S n T. Thanks in advance. I have got an idea but an not totally sure

2. I assume this is to be solved numerically.

(S intersect T) is a subspace, W. Thus W=perp(perp(W)).

perp(W)=(perp(S) union perp(T))

To annihilate the last set, a vector must annihilate the basis for perp(S) and the basis for perp(T). In other words, it must lie in the null space of the matrix containing these (4) vectors. This null space is one dimensional, being generated by the vector

-0.5477
-0.4472
0.2236
0.6708

Obtained from Matlab by the command:

null([null([ 1 1 1 0; 0 -1 1 0]) null([1 0 0 1; 0 2 0 -1])])

3. If it didn't need to be solved arithmetically and instead each vector was assigned a value E.g a(1,1,1,0)^t would it eventually lead to a=c=d=-d. Thanks