Math 317: Linear Algebra
Homework 6
Due: March 7, 2016
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 3.1: 1–5,7,10,12,13
Section 3.2: 2,3,4,13
Turn in the following problems.
1. For each of the following sets V , determine if V forms a subspace in Rn . Justify your
answer in each case by providing a proof or a counterexample where appropriate.
(
)
n
X
(a) V = x ∈ Rn :
xi = 0
i=1
(b) V = x ∈ R : Ax = b, A 6= 0, b 6= 0, b ∈ Rm , A ∈ Rm×n
n
2. Determine which of the following subsets of Rn×n are in fact subspaces of Rn×n .
Justify your answer in each case by providing a proof or a counterexample where
appropriate.
(a) The symmetric matrices.
(b) The nonsingular matrices.
(c) The singular matrices.
3. Section 3.1, Problem 6
4. Section 3.1, Problem 8
5. Section 3.1, Problem 11
6. Section 3.1, Problem 18
7. Section 3.2, Problem 3b
8. Section 3.2, Problem 6a
9. Section 3.2, Problem 10
1