1. Prove or give a counterexample to the following claim:
Claim: Let V be a vector space over the field F and suppose that W1, W2 and W3
are subspaces of V such that W1 + W3 = W2 + W3. Then W1 = W2.
2. Consider the following subspaces of the vector space R^3
over the field R of real numbers:
subspace U1, which is the plane x + y + z = 0 and subspace U2, which is the yz-plane.
a) Can R^3 be written as a sum of U1 and U2? Justify your answer.
b) Can R^3 be written as a direct sum of U1 and U2? Justify your answer.
Here x, y and z denote the usual Cartesian coordinates.
3. Let V be a vector space over the field F and suppose (v1, v2, ... , vn) is
a linearly independent set of vectors in V . Now suppose there exists w in V such
that (v1 + w, v2 + w, ... , vn + w) is a linearly dependent set of vectors in V . Prove
that w in span(v1, v2, ... , vn).