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
Math 3130 (4) - Linear Algebra
Solutions to Exam 2
1) (20 pts.) The matrix A below is known to be invertible. Compute its inverse using the procedure outlined in class and then use it to solve the linear system Ax = b where b is given below.
2 6 6
1
PRACTICE FINAL EXAM
1) If possible, nd a general solution to the system below by using row reduction on the associated
augmented matrix. You must at least reduce the matrix to Echelon form before reading o the
solution.
y+z=1
2x + 6y = 2
x + y + 2z = 3
SOLUTIONS TO HOMEWORK III
1)
a) In order to nd the inverse of A, we perform the elementary row operations needed to reduce A to Reduced
Echelon form on the extended matrix [A | I ]. If the reduced Echelon form of A turns out to be the identity matrix, the
TEST 3 SOLUTIONS
1) (20 pts.) If possible, diagonalize the matrix
1 1 2
A= 0 0 0
0 1 2
.
Sol. The characteristic equation for A is
(1 )()(2 ) = 0,
which yields the eigenvalues
1 = 1,
2 = 0,
3 = 2.
For 1 = 1 we solve (A 1 I)x = 0 and nd eigenvectors of th
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 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 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
SOLUTIONS TO HOMEWORK II
1)
a) Notice that u2 is 5 times u1 . Since there are only two vectors and they are multiples of each other, then they
are linearly dependent.
b) Since there are more vectors (3 vectors) than there are entries in each vector (2 ent
Math 3130 - Linear Algebra
Solutions Exam 1
1) (10 pts.) Find the general solution to the system below by using row reduction on the
associated augmented matrix. You must at least reduce the matrix to Echelon form before reading
o the solution.
2x1 + x3 =
HOMEWORK VI SOLUTIONS
1)
The position
x1 (t)
x2 (t)
x(t) =
of a particle is initially
3
2
x(0) =
,
and, for t > 0, satises
x1 (t) = 2x1 (t) 5x2 (t),
(0.1)
x2 (t) = x1 (t) + 4x2 (t).
Find its position at t = 5 seconds.
Sol. We may write the system in matri