MATH 300
Linear Algebra I
Lecture 4
September 1, 2010
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MATH 300 Linear Algebra I Lecture 4
Section 1.5: Solution Sets of Linear Systems
Denitions
A system of linear equations is said to be homogeneous if it can
be written in the form Ax = 0, wher
MATH 300
Linear Algebra I
Lecture 8
September 20, 2010
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MATH 300 Linear Algebra I Lecture 8
Section 1.9: The Matrix of a Linear Transformation
Linear Transformations as Matrix Transformations
Generally, an explicit formula for T (x) is desired ra
MATH 300
Linear Algebra I
Section 4.1 Lecture
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MATH 300 Linear Algebra I Section 4.1 Lecture
Section 4.1: Vector Spaces and Subspaces
General Ideas of the Section
Diverse mathematical systems share many properties with
Rn .
Vector spaces are coll
MATH 300
Linear Algebra I
Section 5.1 Lecture
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MATH 300 Linear Algebra I Section 5.1 Lecture
Section 5.1: Eigenvectors and Eigenvalues
Denitions
An eigenvector of an n n matrix A is a non-zero vector x
such that Ax = x for some scalar . A scalar
MATH 300
Linear Algebra I
Section 2.9 Lecture
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MATH 300 Linear Algebra I Section 2.9 Lecture
Section 2.9: Dimension and Rank
Purpose of selecting a basis
Why do we care about a basis of H rather than any set that
spans H:
Each vector in H is a un
MATH 300
Linear Algebra I
Section 5.2 Lecture
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MATH 300 Linear Algebra I Section 5.2 Lecture
Section 5.2: The Characteristic Equation
Theorem
The Invertible Matrix Theorem (continued)
Let A be an n n matrix. Then A is invertible iff:
s. The numbe
MATH 300
Linear Algebra I
Section 5.3 Lecture
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MATH 300 Linear Algebra I Section 5.3 Lecture
Section 5.3: Diagonalization
Matrix factorization
Recall that A is said to be similar to B if there exists an
invertible matrix P such that
A = PBP 1
Thi
MATH 300
Linear Algebra I
Section 6.2 Lecture
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MATH 300 Linear Algebra I Section 6.2 Lecture
Section 6.2: Orthogonal Sets
Denition
A set of vectors u1 , u2 , . . . , up in Rn is said to be an orthogonal set if each pair of
distinct vectors from t
MATH 300
Linear Algebra I
Section 6.1 Lecture
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MATH 300 Linear Algebra I Section 6.1 Lecture
Section 6.1: Inner Product, Length, and Orthogonality
What is distance?
How far is the vector
1
1
1
1
1
from the vector
3
3
3
3
3
?
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MATH 300
MATH 300
Linear Algebra I
Section 5.4 Lecture
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MATH 300 Linear Algebra I Section 5.4 Lecture
Section 5.4: Eigenvectors and Linear Transformations
Goals of this section
Place factorization A = PDP 1 in the context linear
transformation.
Extend und
Course Syllabus
MATH 300 Linear Algebra I
Fall 2010
Instructor: Ian Besse, Ph.D.
Email: [email protected]
Lecture Times/Location:
M/W: 5:30 6:45 PM in 309 Haag Hall
Text: Linear Algebra and Its Applications,
Updated 3rd Edition, David C. Lay
Office: 205C Ma
MATH 300
Linear Algebra I
Section 3.2 Lecture
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MATH 300 Linear Algebra I Section 3.2 Lecture
Section 3.2: Properties of Determinants
Theorem
Row Operations
Let A be a square matrix.
If a multiple of one row of A is added to another row to
produce
MATH 300
Linear Algebra I
Section 3.1 Lecture
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MATH 300 Linear Algebra I Section 3.1 Lecture
Section 3.1: Introduction to Determinants
Determinants of n n matrices
Recall 2 2 matrix A is invertible iff det A = 0.
True also for n n matrices.
Need
MATH 300
Linear Algebra I
Lecture 5
September 8, 2010
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MATH 300 Linear Algebra I Lecture 5
Section 1.6: Applications of Linear Systems
Application to Economics: A Simple Economic Model
Suppose a nations economy is divided into many sectors
corres
MATH 300
Linear Algebra I
Lecture 6
September 13, 2010
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MATH 300 Linear Algebra I Lecture 6
Section 1.7: Linear Independence
Denitions
An indexed set of vectors cfw_v1 , . . . , vp Rn is said to be
linearly independent if the vector equation
x1
MATH 300
Linear Algebra I
Lecture 1
August 23th, 2010
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MATH 300 Linear Algebra I Lecture 1
Section 1.1: Systems of Linear Equations
Denition
A system of linear equations (or linear system) is a collection
of one or more linear equations involving
MATH 300
Linear Algebra I
Lecture 3
August 30, 2010
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MATH 300 Linear Algebra I Lecture 3
Section 1.3: Vector Equations
Denition
A matrix with only one column is called a column vector (or simply
a vector).
Examples
1
3
2
1
0
8
4
0
1
0
2
7
MATH 300
Linear Algebra I
Lecture 7
September 15, 2010
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MATH 300 Linear Algebra I Lecture 7
Section 1.8: Introduction to Linear Transformations
A dierent perspective on Ax = b:
Instead of thinking about b as a linear combination of the columns
of
MATH 300
Linear Algebra I
Section 2.2 Lecture
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MATH 300 Linear Algebra I Section 2.2 Lecture
Section 2.2: The Inverse of a Matrix
The Multiplicative Inverse
Real numbers
41 4 = 1 = 4 41
We call the real number 1, the multiplicative identity for t
MATH 300
Linear Algebra I
Section 2.3 Lecture
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MATH 300 Linear Algebra I Section 2.3 Lecture
Section 2.3: Characterizations of Invertible Matrices
The Invertible Matrix Theorem
Let A be an n n matrix. The the following statements are equivalent:
MATH 300
Linear Algebra I
Lecture 9
September 27, 2010
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MATH 300 Linear Algebra I Lecture 9
Section 2.1: Matrix Operations
Notation
An m n matrix A as the concatenation of column vectors:
A = [a1 , a2 , . . . , an ]
Subscript ordered pairs denote
MATH 300
Linear Algebra I
Section 2.4 Lecture
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MATH 300 Linear Algebra I Section 2.4 Lecture
Section 2.4: Partitioned Matrices
Denition
A partitioned (or block) matrix is one whose entries are
themselves submatrices (often called blocks).
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MATH 300
Linear Algebra I
Section 2.8 Lecture
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MATH 300 Linear Algebra I Section 2.8 Lecture
Section 2.8: Subspaces of Rn
Denition
A subspace of Rn is any set H Rn that has these three
properties:
1
The zero vector is in H
2
For each u and v in H
MATH 300
Linear Algebra I
Lecture 2
August 25th, 2010
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MATH 300 Linear Algebra I Lecture 2
Section 1.2: Row Reduction and Echelon Forms
Denition
A leading entry of a row refers to the leftmost nonzero entry (in a
nonzero row).
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MATH 300