Question

You are given two finite dimensional subspaces of some inner product space. If one subspace is of lower dimension than the other, show whether there must exsist atleast one nonzero vector in the larger space that is orthogonal to all vectors in the smaller space.

Answer 1

We can suppose that U,V are subspaces of some space X. dim(U)<dim(V).
If
we consider their intersection W, then choosing base w_1,...,w_k in W, be can
compeate it to the base of U by vectors u_1,...,u_m.
Then we can choose any
vector v_0 from V, that is not linear combination of vectors from W. Then

v_0, w_1,...,w_k, u_1,...,u_m are linearly independent, and vector v_0 can
be ortogonalized to all other these vectors, thus
it will be orthogonal to U.