Isomorphism and Subspace Intersection in Complex Vector Space

[cocobaby]

Homework Statement

Let V be a vector space over the field of complex numbers, and suppose there is an isomorphism T of V onto C3. Let a1, a2, a3,a4 be vectors in V such that

Ta1 = (1, 0 ,i)
Ta2 = (-2, 1+i, 0)
Ta3 = (-1, 1, 1)
Ta4 = (2^1/2, i, 3)

Let W1 be the suubspace spanned by a1 and a2, and let W2 be the subspace spanned by a3 and a4. What is the intersection of W1 and W2?

Homework Equations

The Attempt at a Solution

In this problem, can we find numerical values for the intersection of the given two subspaces? It's obvious that the intersection would be a line passing through the origin, but given no numerical values of a1,2,3,4 in V, can we find a single vector that spans the line?

 

[Dick]

It's not totally obvious it's a line. How did you show it's a line? If you can do that you can probably figure out how to show the intersection of W1 and W2 is span(v) where you can express v in terms of a linear combination of a1, a2, a3 and a4.