LINEAR ALGEBRA THEORY EXAM 1
This exam will focus primarily on the content of sections 3.13.2, 5.15.5, 6.1, 6.2, and 6.4 of our
textbook. The main topics of study in the second half of this course were
(1) Linear transformations between vector spaces,
(2)
LINEAR ALGEBRA THEORY EXAM 1
The focus of this exam is the content of sections 1.11.9, 2.12.3, and 4.14.7 of our textbook.
1. Vector spaces
Vector space. A vector space V is a nonempty set of objects with two operations: addition and scalar
multiplication
Linear algebra quiz 7
1. (1 pt) Write your name in the top left corner of this page.
5 1
2. (2 pts) Compute the eigenvalues of the matrix A =
.
8 1
answer: Recall that the characteristic polynomial of a 2 2 matrix is
2 tr(A) + det(A),
so for this particul
Linear algebra quiz 5
1. (1 pt) Write your name in the top left corner of this page.
2. Let T : P2 R3 be the function dened by
p(1)
T (p(x) = p(0) .
p(1)
(a) (4 pts) Show that T is a linear transformation.
(b) (6 pts) Find the matrix for T with respect to
Linear algebra quiz 4
1. (1 pt) Write your name in the top left corner of this page.
2. (3 pts) Suppose A is a 3 5 matrix. List all possibilities for the rank and nullity of A. (Be clear as to
which values are ranks and which are nullities.)
answer: The r
LINEAR ALGEBRA PRACTICE EXAM 4
(1) Diagonalization. Let
2
2 1
3 1 .
A= 1
1 2 2
If A is diagonalizable, nd a diagonal matrix D that is similar to A.
answer: We begin by computing the eigenvalues of A. A few row operations help simply the
computation.
2
2
1
LINEAR ALGEBRA PRACTICE EXAM 2
(1) Invertible matrices.
Suppose A, B, and X are matrices that satisfy the relation AX
0
1 0
1 3
2
A = 1 3 1 and B = 1 2
3
3
4 5
1 2 0
A = B, where
4
0 .
6
Solve for X.
answer: Solving for X symbolically, we see that
AX = B
LINEAR ALGEBRA PRACTICE EXAM 2
(1) Matrices for linear transformations between general vector spaces. Let T : P2 R3 be
dened by
p(0)
T (p(x) = p(1) .
p(2)
Let B = cfw_2, 1 + t, t2 be a basis for P2 and let
1
0
1
D = 0 , 1 , 0
0
1
1
be a basis for R3
LINEAR ALGEBRA PRACTICE EXAM 1
(1) Matrix algebra.
4
0
1
3 1 2 0
0
1 1
0 , B = 6 0
1 1, and C = 2
3
4 . Compute (A+C)T B.
Let A = 1 3
2 1 2
0
5 10 0
1 2 3
(2) Systems of linear equations.
Give a parametric solution to the following system of linear equati
MATRIX ALGEBRA I
6
Exercise 1. Let A =
2
(1)
(2)
(3)
(4)
0
2
1
4
and B = 0
0
2
3
2 . Compute each of the following.
1
AB
BA
(AB)T B T AT
3
B 2 AT
4
9
2
1
Exercise 2. Suppose A
= 2 and A
= 0. What is A?
1
0
6
2
Vocabulary: Dene each of the following te
TEST 1 - REVIEW (WITH SOLUTIONS)
Every answer must be justied mathematically or, when applicable, by citing the appropriate
Theorem or result discussed in class.
1) Solve using row reduction on the associated augmented matrix. You must at least get to
Ech