Math 211 - Linear Algebra
Material for Midterm 2
The following sections/subsections will be covered on Midterm 2. Apart
from these, the examples done in class, the quizzes, and the assigned homework are all important.
Chapter 2, Sections 2.1, 2.2, 2.3.
Quiz 0 Solutions, Math 211, Section 1 (Vinroot)
Consider the following linear system of equations, which geometrically represents three planes:
x1 2x2 + 2x3 =
2
= 3
x1 + 2x2
2x1 4x2 + 2x3 =
1
Use an augmented matrix and the row reduction algorithm to dete
Quiz 1 Solutions, Math 211, Section 1 (Vinroot)
Consider the following three vectors in R3 :
1
3
1
v1 = 2 , v2 = 5 , v3 = 1
2
6
1
.
Determine, with an explanation, whether every vector in R3 is a linear combination of v1 , v2 , and
v3 . Is Spancfw_v1 , v2
Quiz 3 Solutions, Math 211, Section 1 (Vinroot)
(a): If A =
1
0
0 2
and B =
0 1
Solution: 2A 3B = 2
1
2 0
0
0 1
3
, compute 2A 3B.
0 2
2 0
=
2
0
0 2
0 6
6 0
=
2
6
6 2
.
(b): If B is a 3 7 matrix, and A is a 3 5 matrix, what is the size is AT B? Explain br
Quiz 4, Math 211, Section 1 (Vinroot)
Name:
(a): If V is a vector space, give the conditions which dene a subset W to be a subspace of
V.
Solution:
W is a subspace of V if: (i) the zero vector 0 is in W , (ii) whenever u and v are
in W , then u + v is in
Quiz 2, Math 211, Section 1 (Vinroot)
Name:
Suppose that T : R2 R3 is a linear transformation such that
3
5
1
0
T
= 2 and T
= 2 .
1
1
0
2
(a): Using the information given and the fact that T is a linear transformation, nd T
1
0
,
and explain.
Soluti
Quiz 5 Solutions, Math 211, Section 1 (Vinroot)
(a): Given that B =
2
1
,
4
3
is a basis for R2 , nd the matrix C which satises [v]B = Cv
2
for any v in R .
Solution:
We know that for any v in R2 , we have PB [v]B = v where PB =
2 4
. Then
1 3
1
[v]B = PB