Math 1152 Spring 2014
1.4
23
Gaussian Elimination
Definition 1.4.1. Gauss-Jordan elimination is the systematic procedure to bring a
matrix to reduced row echelon form by starting at the upper left and working down to the
lower right using only elementary

Math 1152 Spring 2013
5.3.1
121
Systems of Dierential Equations
Denition 5.3.9. A dierential equation, is an equation with the derivative(s) of an
unknown function.
Example 5.3.10. The following are dierential equations:
dy
dy
a) dx = 2x
b) dx = 2y
c) t2

Math 1152 Spring 2013
6.5
153
Orthogonal Matrices and Operators
Denition 6.5.1. A n n matrix A is an orthogonal matrix if its columns form an
orthonormal basis for Rn .
A linear operator on Rn is called an orthogonal operator if its standard matrix is an

Chapter 5
Eigenvalues, Eigenvectors, and
Diagonalization
5.1
Eigenvalues and Eigenvectors
Graphs are an important tool for understanding functions. Unfortunately, even for the
smallest cases of linear transformations, graphs would require 4 dimensional gr

Math 1152 Spring 2013
5.1.1
103
Complex Numbers
Denition 5.1.11. The set of Complex Numbers, denoted C, is the set of all ordered
pairs (a, b), where a and b are real numbers. To consider them numbers, we dene addition
and multiplication by:
(a, b) + (c,

Math 1152 Spring 2013
4.5
95
Matrix Representation of Linear Operators
Recall that a linear operator is a linear transformation T : Rn Rn . Back in section 2.7,
we discussed the matrix of a linear operator with respect to the standard basis. It had the
pr

Math 1152 Spring 2013
5.3
115
Diagonalization of Matrices
Denition 5.3.1. To diagonalize a matrix A means to nd a matrix P such that P 1 AP =
D is a diagonal matrix.
A matrix A is diagonalizable if there exists such a matrix P .
Example 5.3.2. Show that A

Math 1152 Spring 2013
5.2
The Characteristic Polynomial
Example 5.2.1. Find the eigenvalues and eigenvectors of:
6 10 28
A = 2 3 4
1
2 7
111
112
The Characteristic Polynomial
Denition 5.2.2. The characteristic polynomial of A is det(A I).
Example 5.2.3.

Math 1152 Spring 2013
4.4
91
Coordinate Systems
Denition 4.4.1. A coordinate system of a vector space V is an ordered basis B
Any basis of a vector space becomes an ordered basis if you give it an order.
Example 4.4.2. E = cfw_i, j is an ordered basis of

Math 1152 Spring 2013
6.2
147
Orthogonal Vectors
Denition 6.2.1. A subset of Rn is called an orthogonal set if every pair of distinct
vectors in the set is orthogonal.
A subset of Rn is called an orthonormal set if it is orthogonal, and also every element

Chapter 6
Orthogonality
6.1
The Geometry of Vectors
In physics, vector quantities are those that have both size and direction. These are both
geometric concepts and this chapter looks further at these two ideas.
3
2
Example 6.1.1. Find the length of u =
a

Math 1152 Spring 2013
6.6
157
Symmetric Matrices
Recall:
Denition 6.6.1. A matrix A is symmetric if AT = A.
Theorem 6.6.2. If u and v are eigenvalues of a symmetric matrix A that correspond to
distinct eigenvalues, then u and v are orthogonal.
Theorem 6.6

Chapter 1
Matrices and Vectors
1.1
Basic Definitions
Definition 1.1.1. A (real) scalar is a real number. We denote the set of real numbers by
R.
Definition 1.1.2. A matrix A is a rectangular array of scalars.
1 0 53
Example 1.1.3. A =
and u = 4.32 1 e are

Math 1152 Spring 2014
1.3
15
Systems of Linear Equations
Definition 1.3.1. A linear equation in the variables x1 , x2 , . . . , xn is an equation which
can be written in the form
a1 x 1 + a2 x 2 + + an x n = b
Each xi is called a
Each ai is called a
Final

Chapter 1
Matrices and Vectors
1.1
Basic Definitions
1.2
Linear Combinations, Matrix-Vector Products
Definition 1.2.1. If cfw_u1 , u2 , . . . , uk is a set of vectors in Rn , then a linear combination
of these vectors is any sum of the form c1 u1 + c2 u2

Math 1152 Spring 2014
1.7
35
Linear Dependence and Linear Independence
Definition 1.7.1. A set S = cfw_v1 , v2 , . . . , vk in Rn is called linearly independent if
c1 v1 + c2 v2 + + ck vk = 0 if and only if c1 = 0, . . . ck = 0
If it is not linearly inde

Math 1152 Spring 2014
1.6
31
Span of a set of Vectors
Definition 1.6.1. The span of a set of vectors S = cfw_u1 , u2 , . . . , uk in Rn is
Span S = cfw_v Rn |v = c1 u1 + c2 u2 + + ck uk , ci R
that is, the set of all linear combinations of vectors in S.

Math 1152 Spring 2014
2.3
49
Invertibility, Elementary Matrices
Definition 2.3.1. Let A and B be n n matrices. We say that B is the inverse of A if
AB = In and BA = In . We write A1 = B.
Example
2.3.2.
Show that B is the
inverse of A, where:
3 7
5 7
A=
,

Math 1152 Spring 2013
6.1.2
141
Lines and Planes
Denition 6.1.35. A direction vector of a line is a vector, v, that is parallel to the line.
Example 6.1.36. Find a direction vector of the line 2x + 5y = 10.
Denition 6.1.37. A vector equation of a line is

Math 1152 Spring 2013
6.1.1
135
Cross Product
Denition 6.1.27. If u and v are in R3 then the cross product, denoted u v is dened
by
u v = u v (sin )n
where is the angle between u and v, and n is a unit normal to both u and v chosen by
the right hand rule.

Math 1152 Spring 2013
4.3
89
The Dimension of Subspaces Associated with a
Matrix
From what we did in the previous section, we have:
Theorem 4.3.1. Suppose that A is an m n matrix. Then
a) dim Row A = rank A.
b) dim Col A = rank A.
c) dim Null A = n rank A

Math 1152 Spring 2013
4.2
83
Basis and Dimension
Recall that back in sections 1.6 and 1.7 we discussed the ideas of span and linear independence. These are connected in the following way:
Theorem 4.2.1. Suppose that r > k and V is any vector space.
If V =

Math 1152 Spring 2013
1.6
31
Span of a set of Vectors
Denition 1.6.1. The span of a set of vectors S = cfw_u1 , u2 , . . . , uk in Rn is
Span S = cfw_v Rn |v = c1 u1 + c2 u2 + + ck uk , ci R
that is, the set of all linear combinations of vectors in S. We

Math 1152 Spring 2013
1.4
23
Gaussian Elimination
Denition 1.4.1. Gauss-Jordan elimination is the systematic procedure to bring a
matrix to reduced row echelon form by starting at the upper left and working down to the
lower right using only elementary ro

Math 1152 Spring 2013
1.2
9
Linear Combinations, Matrix-Vector Products
Denition 1.2.1. If cfw_u1 , u2 , . . . , uk is a set of vectors in Rn , then a linear combination
of these vectors is any sum of the form c1 u1 + c2 u2 + + ck uk . The scalars c1 , c

Math 1152 Spring 2013
1.5
29
Applications
Example 1.5.1. Determine the currents in the following circuit.
e
e e e e
e e e
e
e
e
e
e e e e
e e e
e
e
e
E
6
5
r
r
rr
r
rr
r
rr
r
T
6V
r
r
rr
r
rr
r
rr
r
10
c
c
+
e
e e e e
e e e
e
e
e
11
E
'
'
10 V
5
30
Appl

Chapter 1
Matrices and Vectors
1.1
Basic Denitions
Example 1.1.1. A small company sells widgets. In December they sold 5 deluxe blue
widgets.
The number 5 in the above example is called a scalar quantity.
Denition 1.1.2. A (real) scalar is a real number.

Math 1152-S10
Assignment #3
April 4, 2013
This assignment is due by 2:00 PM, Monday, April 15, 2013. Late assignments will not be
accepted under any circumstances and will be given a score of zero.
All calculations are to be done using Maple software. Han

Index
basis, 84
l
inearl independent, 35
y
codom ain, 57
cof
actor, 70
col n space, 81
um
C om pl num bers, 1
ex
com position, 65
coordinate system , 91
coordinates, 91
m atrix
augm ented, 19
cof
actor, 70
col n space, 81
um
denition, 2
determ inant, 70
d

Assignment 2
This assignment is due Thursday March 21 at 2pm.
Before you start on this assignment, you must work through the MAPLE worksheet called
Trusses.mws, which is on the course website.
Important: You must follow the following instructions. After M