Linear Algebra Answers

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.

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.