Linear Algebra Answers

Let u and v be two non-zero N-dimensional complex column vectors. Show that the rank of the N by N matrix uv^(conjugate transpose) is one.

Show that rank(A+B) is never greater than the sum of rank(A) and rank(B).

Let A and B be M by N matrices, P an invertible M by M matrix, and Q an invertible N by N matrix, such that B = PAQ, that is, the matrices A and B are equivalent. Show that the rank of B is the same as the rank of A. (Show that A and AQ have the same rank).

Let A be an M by N matrix. When does A have a left inverse? When does it have a right inverse?

Prove that a square matrix is invertible if and only if it has a full rank.

Show that:
a) If U = {u^1, u^2,...,u^N} is a spanning set for W, then U is a basis for W iff, after the removal of any one member, U is no longer a spanning set for W; and

b) If U = {u^1, u^2,...,u^N} is a linearly independent set in W, then U is a basis for W iff, after including in U any new member from W, U is no longer linearly independent.

Decide whether the set of all real functions form a vector space.

Geometrically describe the fundamental vector subspaces of a transformation.

How do the “span” and “basis” of a subspace differ?

Prove that every finite dimensional vector space has a basis.