MA 265 Lecture 23
Section 4.9
Rank of a Matrix
Definition Let
A=
.
.
a1n
a2n
.
.
am1 am2
amn
a11
a21
.
.
a12
a22
.
.
be an m n matrix.
The rows of A,
The columns of A,
Remark If A and B are row equivalent matrices, then
1
We can use this remark to find
MA 265 Lecture 12
Section 4.1
Vectors in the Plane and in 3-Space
Definitions of scalar and vector
Measurable quantities that can be completely described by giving their magnitude are
called
For example,
.
Measurable quantities that require for descript
/
Fall 2008
MA 265
FINAL EXAM
~anae
_
PUID# _
Instructor _
Section# _ Class Tinae _
INSTRUCTIONS
1. Make sure you have a complete test. There are 12 different test pages, including this cover
page.
2. Your PUID# is your student identification number. DO N
MATH 265 FINAL EXAM Name and ID: Instructor: Section or class time:
Instructions: Calculators are not allowed. There are 8 problems in the first part worth 13 points each. There are 12 problems in the second part worth 8 points each. The total is 200 poin
MATH 265 PRACTICE FINAL EXAM
Part I 1. Consider a linear system whose augmented matrix is of the form 1 1 3 2 1 2 4 3 1 3 a b (a) For what values of a and b will the system have infinitely many solutions? (b) For what values of a and b will the system be
Math 265, Spring 2002
Solution for Final Sample Problems
1. (a) The rank of A is 3 since there are three leading ones in rrefA. (b) The nullity of A is 2. nullity + rank = # of columns = nullity + 3 = 5 = nullity = 2. Or, nullity = # of columns without le
MATH 265 FINAL EXAM, Spring 2007
Name and ID:
Instructor:
Section or class time:
Instructions: Calculators are not allowed. There are 25 multiple choice problems
worth 8 points each, for a total of 200 points.
1
14
2
15
3
16
4
17
5
18
6
19
7
20
8
21
9
22
MA 265 Lecture 7
Section 2.3
Elementary Matrices; Finding A1
Definition
An n n elementary matrix of type I, II, or III is
Example 1. The following are elementary matrices
0 0 1
1 0 0
E1 = 0 1 0 , E2 = 0 3 0 ,
1 0 0
0 0 1
1 2 0
E3 = 0 1 0 .
0 0 1
Theorem L
MA 265 Lecture 13
Section 4.2
Vector Spaces
Definition of Vector Space
A
and
is a set V of elements on which we have two operations
defined with following properties:
(a) If u and v are any elements in V , then u v is in V .
(V is closed under the operati
MA 265 Lecture 8
Section 3.1
Definition of Determinants
Definition of Permutation
Let S = cfw_1, 2, , n be the set of integers from 1 to n, arranged in ascending order.
We can consider a permutation of S to be a one-to-one mapping of S to itself. For exam
MA 265 Lecture 15
Section 4.3
Subspaces (cont)
Definition of Linear Combination
Let v1 , v2 , , vk be vectors in a vector space V .
Example 1. Every polynomial of degree 2 is a linear combination of t2 , t, 1.
Example 2. Show
1
nation of v1 = 0
1
a
that t
MA 265 Lecture 10
Section 3.3
Cofactor Expansion
Definition of Minor
Let A = [aij ] be an n n matrix.
Definition of Cofactor
Let A = [aij ] be an n n matrix.
Example 1. Find the cofactors A12 and A23 if
4 3 2
A = 4 2 5
2 4 6
1
Theorem (cofactor expansion
MA 265 Lecture 14
Section 4.3
Subspaces
Definition of Subspaces
Let V be a vector space and W a nonempty subset of V . If
To check W is a subspace of V , we only need to check closure property for and on W :
Theorem Let V be a vector space with operation
MA 265 Lecture 5
Section 2.1
Echelon Form of a Matrix
Definitions
An m n matrix A is said to be in
if
(a)
(b)
(c)
(d)
An mn matrix satisfying properties (a), (b), (c) is said to be in
We can define
ilar manner.
and
Example 1. Determine whether the followi
MA 265 Lecture 6
Section 2.2
Solving Linear Systems
Example 1. Solve the linear system whose augmented matrix has the echelon form:
1 2 0 3
0 1 1 2
0 0 1 1
To solve a linear system Ax = b,
1. form the augmented matrix
2. transform to a matrix [C | d] in
MA 265 Lecture 20
Section 4.6
Basis and Dimension
Definition of Basis
The vectors v1 , v2 , , vk in a vector space V are said to form a basis for V if
Remark
1
0
0
3
Example 1. In R , the vectors 0 , 1 , 0 , form a basis.
0
0
1
Remark
1
Example 2. Show
MA 265 Lecture 9
Section 3.2
Properties of Determinants
Properties of determinants
Let A and B be n n matrices. Their determinants have following properties
1. det(AT ) =
2. If B = Ari rj or B = Aci cj then
3. If two rows (columns) of A are equal, then
4.
M~th
Final Exam
265 Linear Algebra
St_u_d_en_t_N_a_m_e (p_r_in_t_):_
_
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
Spring 2001
~11
S_tu_d_e_n_t_ID_:_
L _
Circle the name of your instructor (with the time of your class):
Ban
De la Cruz (9:00)
Gottlieb (9:00)
Gottlieb (
Math 265, Midterm 1
Sept 28, 2012
Name:
This exam consists of 8 pages including this front page.
Ground Rules
1. No calculator is allowed.
2. Show your work for every problem unless otherwise stated.
3. You may use one 4-by-6 index card, both sides.
Score
66
CHAPTER 2. DETERMINANTS
2.2
Properties of Determinants
In this section, we will study properties determinants have and we will see how
these properties can help in computing the determinant of a matrix. We will
also see how these properties can give us
122
4.2
4.2.1
CHAPTER 4. VECTOR SPACES
Subspaces
Denitions and Examples
Often, we work with vector spaces which consists of an appropriate subset of
vectors from a larger vector space. We might expect that most of the properties
of the larger space would
Chapter 1
Systems of Linear
Equations and Matrices
1.1
Introduction to Systems of Linear Equations
In this section, we give the main denitions of the concepts studied in this
chapter. In particular, we explain what a system of linear equations is and we
g
40
CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES
1.5
1.5.1
Elementary Matrices
Denitions and Examples
The transformations we perform on a system or on the corresponding augmented
matrix, when we attempt to solve the system, can be simulated by matri
1.2. SOLVING A SYSTEM OF LINEAR EQUATIONS
1.2
1.2.1
7
Solving a System of Linear Equations
Simple Systems - Basic Denitions
As noticed above, the general form of a linear system of m equations in n
variables is of the form
8
a11 x1 + a12 x2 + : + a1n xn =
4.3. LINEAR COMBINATIONS AND SPANNING SETS
4.3
125
Linear Combinations and Spanning Sets
In the previous section, we looked at conditions under which a subset W of a
vector space V was itself a vector space. In the next three section, we look at
the follo
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
The Gram-Schmidt Algorithm
In any inner product space, we can choose the basis in which to work. It often greatly simplies
calculations to wor
Practice Midterm 1
Name:
PUID:
MA 265
No calculators may be used on this exam.
Problem
Possible Points
Number
Points Earned
I
60pts
II
20pts
III
20pts
Total Points 100pts
1
Spring 2017
I
Answer the following questions by clearly circling your answers. No
MA 265 Lecture 19
Section 4.5
Linear Independence
a
Recall The set W of all vectors of the form b is a subspace of R3 .
a+b
Example 1. Show that each of the following sets is a
0
1
S1 = 0 , 1 ,
S2 =
1
1
1
spanning set for W
1
0
3
0 , 1 , 2
1
1
5
MA 265 Lecture 39
Section 8.4
Differential Equations
In this section, we consider the first order differential equations.
A simple example of a differential equation is
d
x(t) = 2x(t).
dt
The first order homogeneous system of differential equations is
x01
MA 265 Lecture 27
Section 5.4
Gram-Schmidt Process
In this section, we introduce a method to obtain an orthonormal basis for a finite dimensional
inner product space, which is called Gram-Schmidt Process.
Question: Why do we want an orthonormal basis?
Exa
MA 265 Lecture 28
Section 5.5
Orthogonal Complements
Definition. Let W be a subspace of an inner product space V .
A vector u in V is said to be orthogonal to W if
The set of all vectors in V that are orthogonal to all vectors in W is called
of W in V ,
MA 265 Lecture 35
Section 7.2 Diagonalization and Similar Matrices (contd)
Recall
We say A and B are similar if
We say A is diagonalizable if
An n n matrix A is diagonalizable if and only if
If D = P 1 AP , how to find D and P ?
Example 1. Find a nons