Linear Algebra Answers

Prove that the following are equivalent:

a) The set U = {u^1, u^2,..., u^n} is linearly independent;
b) u^1 does not equal 0 and no u^n is a linear combination of the members of U that precede it in the list;
c) no u^n is a linear combination of the other members of U.

Let V be a finite dimensional vector space and W a subspace of V. Prove that W is also finite dimensional.

Prove that If A is a square and there exist matrices B and C such that AB = I and CA = I, then B = C and A is invertible.

Prove that if AB = BA for every N by N matrix A, then B = cI, for some constant c.

Let D be a diagonal matrix such that D_mm does not equal D_nn if m does not equal n. Show that if BD = DB then B is a diagonal matrix.

Let C = AB. Show that Ctranspose = BtransposeAtranspose.

Show that, for each k = 1,...,K, Col_k(C), the kth column of the matrix C = AB, is

Col_k(C) = ACol_k(B).

Show that, in the vector space V = R2, the subset of all vectors whose entries sum to zero is a subspace, but the subset of all vectors whose entries sum to one is not a subspace.

Prove that 0v = 0 for all v in V, and use this to prove that (-1)v = -v for all v in V.

How is the value of the determinant related to whether a matrix is singular or non-singular?