SOLUTIONS TO THE TRAINING EXERCISES
FOR THE FINAL EXAM
(1) Let V be the space of polynomials of degree at most n. Consider the
application L(P (X) = X(P (X + 1) P (X). Show that this application is
a well-defined application from V into V . Find its matri
F15 Exam 2 Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
Name
Uni
[1] Find a basis for the subspace V of R4 spanned by the vectors
(1, 1, 0, 1)
(1, 0, 1, 1)
(2, 1, 1, 2)
(3, 2, 1, 3)
(3, 1, 2, 3)
(4, 2, 2, 4)
Extend this basis to a basis for R
F15 Exam 1 Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
Name
Uni
[1] Solve the following system of equations.
v
0 1 1 2 3
2
w
1 1 2 3 5 x = 4
1 2 3 5 8 y
6
z
v
w
x =
y
z
F15 Exam 1 Problem 2
Linear Algebra, Dave Bayer
[Reserved f
F14 11:40 Exam 3
Linear Algebra, Dave Bayer
[1] Find the determinant of the matrix
3
1
A=
1
2
3
4
2
2
3
1
1
4
3
1
4
2
[2] Find the inverse of the matrix
1 0 2
A=1 1 1
1 3 2
[3] Using Cramers rule, solve for y in the system of equations
a 1 2
x
1
b 1 3
F14 Practice 3 Problem 1
Linear Algebra, Dave Bayer
[1] Find the determinant of the matrix
1
3
A=
1
0
1
4
1
1
0
1
0
0
2
5
1
1
det(A) =
F14 Practice 3 Problem 2
Linear Algebra, Dave Bayer
[2] Find the determinant of the matrix
2
1
A=
1
1
1
1
3
1
1
1
0
0
3
F14 11:40 Exam 2
Linear Algebra, Dave Bayer
[1] Find the 3 3 matrix which maps the vector (1, 1, 1) to (2, 2, 2), and maps each point on the plane
x + y + z = 0 to itself.
[2] Find a basis for the row space and a basis for the column space of the matrix
1
F14 Practice 1
Linear Algebra, Dave Bayer
[1] Solve the following system of equations.
w
6 1 8 1
x = 4
4 0 5 1 y
3
z
[2] Using matrix multiplication, count the number of paths of length ten from x to z.
x
y
z
[3] Express A as a product of elementary m
F15 Exam 3
Linear Algebra, Dave Bayer
[1] Find the determinant of the matrix
3
1
A=
1
1
1
6
3
1
1
1
6
6
3
1
1
1
1
1
1
1
1
1
1
6
3
[2] Using Cramers rule, solve for z in the system of equations
a 2 1
x
3
b 3 1 y = 1
c 1 1
z
2
[3] Let f(n) be the determ
3 3 Exercise Set M (recurrence relations), November 27, 2016
3 3 Exercise Set M (recurrence relations)
Linear Algebra, Dave Bayer, November 27, 2016
[1] Find An where A is the matrix
0
1
0
1 1
A = 1
1 1
1
3 2 1
1 0 1
0
1 1
n
(1)
1
2
3
2
1 + 1 0 1 +
0
2
PRACTICE MIDTERM II SOLUTIONS
by
Mirela Ciperiani
Problem 1
Compute the determinants of the following matrices and state if the
matrix is singular or not.
A=
1
0
C=
0
0
3
1
0
0
5 7
3 2
7 2
1 2
1 2
3 1
2 1 3
B = 0 3 2
0 2 1
1 3
7
2
0
1
1
2
D=
0
0
1 2
2 6 1
SOLUTIONS TO THE TRAINING EXERCISES
FOR THE SECOND MIDTERM
(1) Solve the system Ax = b using Cramers rule where
1
1 2 3
A = 2 3 1 , b = 1
1
3 2 1
Amswer : The solution is x = y = z = 1/6.
(2) For each of the following sets of elements in M2 (R) (the ve
F16 10:10 Exam 1 Problem 1
Linear Algebra, Dave Bayer
Test 1
Name
[Reserved for Score]
8test1b1p1
6 9 1
Uni
[1] Find the general solution to the following system of equations.
w
5 7 1 2
3
3 4 1 1 x = 2
y
2 3 0 1
1
z
w
x
=
y
z
F16 10:10 Exam 1
3 3 Exercise Set M (recurrence relations), November 27, 2016
3 3 Exercise Set M (recurrence relations)
Linear Algebra, Dave Bayer, November 27, 2016
[1] Find An where A is the matrix
0
1
0
1 1
A = 1
1 1
1
[2] Find An where A is the matrix
0
1
0
A = 1 4 1
F16 10:10 Exam 1 Problem 1
[Reserved for Score]
IIIII III
Linear Algebra, Dave Bayer
testlaZpl
Test 1
Name.
Uni.
4"
[1] Find the determinant of the matrix
" 11 5
'G
A
f-
-h
=
1
1
1
1
1
0
3
0
0
0
111 3
1
de(r(:M=
I
S
I
I
M
I
=
11 "
'3.
I
1
1
1
1
4
M
I
I
3
F16 8:40 Exam 1 Problem 1
testlaZpl
Test 1
Name.
[Reserved for Score]
IIIII III
Linear Algebra, Dave Bayer
SQlu-ho\a<;
Uni.
[1] Find the determinant of the matrix
0
0
A
=
1111
13
0
2
3
3
11 5
2
2
5
2
0
2
111 7
J
M
M
I
1
M
0 7T-2-
1 < SS
O 0 M M
M
00 0 G
1
F16 8:40 Exam 1 Problem 1
Linear Algebra, Dave Bayer
Test 1
Name
[Reserved for Score]
2 test1a1p1
0 5 6
Uni
[1] Find the general solution to the following system of equations.
w
2 1 1 3
2
1 1 0 1 x = 3
y
3 2 1 4
5
z
w
x
=
y
z
F16 8:40 Exam 1 P
F14 Practice 4
Linear Algebra, Dave Bayer
[1] Find An where A is the matrix
1 2
1
0
1 3
2
0
A =
[2] Find eAt where A is the matrix
A =
[3] Solve the differential equation y 0 = Ay where
3 1
A =
,
2
0
y(0) =
1
2
[4] Find eAt where A is the matrix
A =
1 2
2
F15 Final Exam Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
9 test1a4p1
8 4 3
Test 1
Name
Uni
[1] Solve the following system of equations.
w
1 1 1 0
1
1 1 1 0 x = 1
y
1 1 1 0
1
z
w
x
=
y
z
w
1
1
0
x
0
1
1
= +
y
0
0 1
z
0
0
0
F14 Homework 4
Linear Algebra, Dave Bayer
[1] Find An where A is the matrix
3 1
2
0
2 3
1
2
A =
[2] Find eAt where A is the matrix
A =
[3] Solve the differential equation y 0 = Ay where
1 3
A =
,
1 3
y(0) =
1
1
[4] Find eAt where A is the matrix
A =
5 4
1
F14 Homework 2 Problem 9
Linear Algebra, Dave Bayer
[9] Let V be the vector space of all polynomials of degree < 2 in the variable x with coefficients in R. Let W
be the subspace consisting of those polynomials f (x) such that f (1) = 0. Find the orthogon
F14 11:40 Exam 1
Linear Algebra, Dave Bayer
[1] Solve the following system of equations.
w
3 1 0 0
3
2 0 1 0 x = 2
y
4 1 0 1
4
z
[2] Using matrix multiplication, count the number of paths of length eight from x to z.
x
y
z
[3] Express A as a product
F14 8:40 Final Exam Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p1
2 2 7 7
Exam 01
Uni
Name
[1] Find the intersection of the following two affine subspaces of R4 .
w
1 0 0 0
x = 1
0 0 0 1 y
1
z
w
1
1 1
x
= 2 + 1 0 r
y
2
F14 8:40 Exam 1
Linear Algebra, Dave Bayer
[1] Solve the following system of equations.
w
1 4 0 1
4
0 2 1 0 x = 2
y
0 2 0 1
3
z
[2] Using matrix multiplication, count the number of paths of length eight from x to z.
x
y
z
[3] Express A as a product
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
2007 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Gove
Practice Exam 1
[1] Solve the following system of equations:
2 1
0
0
1
2 1
0
0 1
2 1
0
0 1
2
w
1
x
= 0
0
y
z
6
[2] Compute a matrix giving the number of walks of length 4 between pairs of vertices of
the following graph:
[3] Express the following ma
E: Additional Practice Problems
Linear Algebra Dave Bayer, May, 2004
Name: Hhsw
Please work only one problem per page, starting with the pages provided, and identify
all continued-ions clearly.
[1] Let A = [ :1 ? ] Write A as CDCl for a diagonal matrix