Math 317: Linear Algebra
Homework 9
The following problems are for additional practice and are not to be turned in: (All
problems come from Linear Algebra: A Geometric Approach, 2nd Edition by ShifrinAdams.)
Exercises: Section 4.2: 1–6, 8, 9
Turn in the following problems.
1. Section 4.2, Problem 2d
2. Section 4.2, Problem 6 (Hint: Find a matrix A so that V = R(A). Finding a basis
for the orthogonal complement of V should now be straightforward.)
3. Section 4.2, Problem 10
4. Section 4.2,

1
1
5. Let A = 
1
0
Problem 11

 
0 −1
1



2 1
1.
and
b
=
1
1 −3
1 1
1
(a) Determine the QR factorization of A.
(b) Use the QR factors from part (a) to determine the least squares solution to
Ax = b.
6. (a) Explain what happens when the Gram-Schmidt process is applied to an orthronormal set of vectors.
(b) Explain what happens when the Gram-Schmidt process is applied to a linearly
dependent set of vectors.
1