MAT 442: Test 1 Solutions
Throughout, F is a eld.
1. Dene the following terms.
(a) Span
If a set S is a subset of a vector space V , then if S = we dene the span of S
to be cfw_O, and otherwise the span of S is the set of all linear combinations of
elemen
MAT 442: Test 1
Instructor: J. Jones
Name:
1
2
3
4
5
6
7
8
Total
15 pts.
10 pts.
20 pts.
10 pts.
15 pts.
20 pts.
10 pts.
10 pts.
110 pts.
Write your answers on separate paper. Make sure your name is on every sheet.
If a problem asks you to prove a result
MAT 442: Test 1 Solutions
Throughout, F is a eld.
1. In each part, say whether the statement is true or false. Then, either give a brief reason
why the statement is true, or a counterexmple showing that it is false.
(a) If V is a vector space over R and u
MAT 442: Test 2 Solutions
Throughout, F is a eld.
1. In each part, say whether the statement is true or false. Then, either give a brief reason
why the statement is true, or a counterexample showing that it is false.
(a) Every square matrix equals a produ
MAT 442: Test 2 Solutions
1. Read these instructions carefully. For each type of row operation give the following:
(a) A description of what it does (so we know which row operation you are talking
about)
(b) The determinant of a corresponding elementary m
MAT 442: Test 3 Solutions
1. For each part, state the denition or theorem for linear operators.
(a) Dene eigenvalue and eigenvector
If T is an operator on a vector space V over F , an eigenvector x V is a non-zero
vector such that T (x) = x for some F . A
Test 1 Solutions
Throughout, F is a eld.
1. Suppose V is a vector space over F . Prove that for all a F ,
a0 = 0
where 0 is the zero vector of V . (This is a theorem from the course.)
For all a F , a0 = a(0 + 0) = a0 + a0. Then subtract a0 from both sides
MAT 442: Test 2 Solutions
Throughout, F is a eld.
1. Read these instructions carefully. For each type of row operation give the following:
(a) A description of what it does (so we know which row operation you are talking
about)
(b) The determinant of a co