MAT 335 Test 1
1. Write the augmented matrix
for the following linear system. Then row reduce the
augmented matrix to nd the
solution set of the system.
2x1 + 7x2 = 1
2x1 + 8x2 + 3x3 = 1
x1 + 4x2 + 2x3 = 5
Sample Problems
1 2 0 5
4. Given A = 4 12 11 4,
1
MAT 335
Review Guide for Test 1
Fall 2013
The test will cover material discussed in class and studied in exercises, from Section 1.1 to
Section 2.3. This guide provides some suggestions and notes some of the most important
denitions, theorems, and skills
Vector equations
vector: a matrix with only one column
o the set of all vectors with 2 entries is denoted R2
R stands for the real numbers that appear as entries in the
vector, and the 2 indicates that each vector contains only
two entries
scalar multiple
Math 335
Homework
1. Balance the chemical equations:
a) B# S$ H# O p H$ BO$ H# S
B
#
!
"
!
S
$
!
!
"
Using the vector representation and solving B" B# B$ B% yields the
H
!
#
$
#
O
!
"
$
!
"
" ! ! $
! " ! #
reduced matrix
! ! " # .
$
! ! !
!
$
#
The so
Math 335
Homework
1. Balance the chemical equations:
a) B# S$ H# O p H$ BO$ H# S
b) MnS As# Cr"! O$& H# SO% p HMnO% AsH$ CrW$ O"# H# O
2. Find the general flow pattern of the network shown in the figure. If all flows are non-negative, what
are the smalles
MAT 335
Extra Assignment
1. Find the general solution to xw Ex, for E
$
"
2.
a) Find the general solution to xw Ex, for E
b) Determine the specific solution if xa!b
#
$
#
"
#
"
B" !
"
%
B# !
3. Find the general solution to xw Ex, for E
-7 10
-4 5
MAT 335
Practice Problems
1. In old growth forests of Douglas fir, the spotted owl feasts mainly upon flying squirrels. If p5
o5
is a
s5
vector describing the population of owls and squirrels (in thousands) at the k>2 month and p5 Ep5" ,
with
%
$
E
% "
MATH 335 Quiz 6
Name:
Show all work. 20 points.
1) Find the equation of the least squares line that best ts (1, 0), (0, 1), (1, 2), (2, 4).
2) Show that if A is an n n symmetric matrix, then (Ax) y = x (Ay) for all x, y Rn .
Review 2 Math 335
Any term in bold face know the denition well enough to state it on the test. The denition you give should be
very similar to the one in the book or one with similar detail.
Section 5.1 eigenvector, eigenvalue, eigenspace, be able to appl
Review for Linear Algebra Comprehensive
All Chapters and Sample Problems refer to Schaums Outline of Linear Algebra, 5th ed., Lipschutz
et al.
Chapter 1: Vectors in Rn and Cn , Spatial Vectors
Theorems: 1.3, 1.4
Sample Problems: 1.3, 1.6, 1.8, 1.10, 1.11,
Review 1 Math 335
Any term in bold face know the denition well enough to state it on the test. The denition you give should be
very similar to the one in the book or one with similar detail.
Section 1.1-1.2 consistent, inconsistent, pivot, general solutio
MATH 335 Quiz 2
Name:
Show all work. 10 points each.
1. Find the standard matrix for the linear transformation T and determine if the transformation is 1-1, onto, both or neither. Justify your answer. T (x1 , x2 , x3 , x4 ) = (x1 + 2x2 , 0, 2x2 +
x4 , x2
MATH 335 Quiz 3
Name:
Show all work. 10 points each. You can check you answers using matlab.
04
by hand and use it to nd eigenvalues
1 4
and eigenvectors of the matrix. What are the dimensions of each eigenspace?
1. Find the characteristic polynomial of A
MATH 335 Quiz 1
Name:
Show all work. 10 points each.
1. Solve the following system of equations. Is the solution unique?
x1 3x3 = 8
2x1 + 2x2 + 9x3 = 7
2x2 + 5x3 = 2
3 6 6
. Describe all solutions of Ax = 0 in parametric vector form.
2 4 2
What geometric
MATH 335 Quiz 4
Name:
Show all work. 20 points .
1) Solve the initial value problem x (t) = Ax(t) for t 0 with x(0) = (3, 2). A =
7 1
33
2) Find the characteristic polynomial, the eigenvalues and a basis for each eigen space for
7 5
13
MATH 335 Quiz 5
Name:
Show all work. 20 points .
1) Let y =
3
8
and u =
. Compute the distance from y to the line through u and the
1
6
origin.
1
3
3
1
1
1
2) Find the closest point to y = in the subspace spanned by v1 = and v2 =
1
1
5
1
1
1
MATH 335 Quiz 7
Name:
Show all work. 20 points .
Assume that a Chicago Taxi company has 1500 cabs in a city and they reside in one of three
areas. OHare Airport, Downtown or Midway airport. Each hour the following transitions
occur:
a) 90% of OHare cabs s
MAT 335
Practice Problems
1. In old growth forests of Douglas fir, the spotted owl feasts mainly upon flying squirrels. If p5
o5
is a
s5
vector describing the population of owls and squirrels (in thousands) at the k>2 month and p5 Ep5" ,
with
E
%
$
% "
Math 335
Practice
1. Balance the chemical equations:
a) Al# O$ C p Al CO#
b) H$ O CaCO$ p H# O Ca CO#
c) PbN' CrMn# O) p Pb$ O% Cr# O$ MnO# NO
2. Find the general flow pattern of the network shown in the figure. If all flows are non-negative, what
is the
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