Homework 3 - due Monday, July 7, in class

Each problem is worth 10 points.

The following problems are from the textbook:

section 3.5: #8, #25
section 3.6: #7, #14 (you may compute A if you want), #28
section 4.3: #27

Additionally, solve the following problems:
1. Let v_1,...v_k be linearly independent vectors in R^n. We have seen in class that k is less than or equal to n. Prove that we can complete this set of vectors to a basis v_1,..., v_n of R^n.
2. Let P_1 be the plane in R^3 spanned by the vectors (1,2,3) and (4,5,6) and P_2 - the one spanned by (7,8,9) and (10,11,12). Describe all vectors in the intersection of these two planes.
3. Let A, B be any matrices such that AB makes sense.
(a) Compare the column spaces of A and AB. Also, compare the nullspaces of AB and B.
(b) Use (a) to show that rank(AB) is less than or equal to rank(A) and rank(B).
4. We found in class an expression for the projection matrix P that projects a vector v onto the column space of a matrix A.
(a) Find the matrix Q that projects a vector onto the left nullspace of A. (Just like P, Q can be expressed in terms of A)
(b) Find the matrices that project a vector onto the rowspace of A and respectively, the nullspace of A.
One easy way to do (b) is to use P and Q from above.