MA 3280 Lecture 01 - Systems of Linear Equations
Monday, January 23, 2014.
Objectives: Review solving systems of linear equations.
Basic Idea
Instead of xs, ys, and zs, well use x1 s, x2 s, and x3 s. This will be more convenient when we start looking
at m
MA 3280 Lecture 15 - Subspaces
Monday, March 3, 2014.
Objectives: Material from 4.5: Subspaces.
Subspaces
Given a vector space V and a subset W V , W is said to be a subspace of V , if W is a vector space on its
own using the same vector addition and scal
MA 3280 Lecture 13 - Vector Spaces
Wednesday, February 26, 2014.
Objectives: Material from 4.1: Introduction to vector spaces.
Vector Spaces
A vector space is an abstract algebraic object that generalizes the structures in an Rn . We will focus on a
speci
MA 3280 Lecture 11 - Orthogonal Vectors
Friday, February 21, 2014.
Objectives: Material from 3.10: Orthogonal vectors.
The angle between vectors u and v is dened to be
(1)
uv
u
v
= cos1
,
or equivalently
(2)
uv
.
u
v
cos() =
Well see that vectors that ar
MA 3280 Lecture 12 - Bases for a Homogeneous System
Monday, February 24, 2014.
Objectives: Material from 3.10: Basis for a solution space to a homogeneous system.
Homogeneous systems
A homogeneous system of linear equations is a system with 0s on the righ
MA 3280 Lecture 10 - Linear Combinations of Vectors
Friday, February 14, 2014.
Objectives: Material from 3.10: Systems of linear equations and linear combinations of vectors and material
from 1.2, 1.3, and 1.4: real vector spaces.
Linear combinations
The
MA 3280 Lecture 09 - Finding Inverse Matrices
Wednesday, February 12, 2014.
Objectives: Justify and formulate a process for nding the inverse of a matrix.
One thing well really want to be able to do is to nd the inverse of a square matrix. Today, Ill want
MA 3280 Lecture 08 - The Matrix Equation
Monday, February 10, 2014.
Objectives: The matrix equation associated with a system of linear equations.
Consider the following equation involving matrices.
2 1 3
x1
2
1 2 3 x2 = 1
(1)
1 2 5
x3
1
If we multiply
MA 3280 Lecture 07 - Matrix Algebra
Friday, February 7, 2014.
Objectives: Review matrix algebra.
You probably have seen some or all of this in your other classes, but Ill just lay out the basic algebraic
operations on matrices.
Matrix addition/subtraction
MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables
Friday, January 31, 2014.
Objectives: Generalize echelon form, and introduce free variables. Material from Section 3.5 starting on
page 64 of the Schaums Outline book.
Consider this example
MA 3280 Lecture 02 - Solution Techniques for Systems of Linear Equations
Friday, January 24, 2014.
Objectives: Solution techniques.
A cute trick
This may not look like anything, but here is one way of solving this system (which well come back to much
late
MA 3280 Lecture 03 - Solution Techniques for Systems of Linear Equations (cont.)
Monday, January 27, 2014.
Objectives: Solution techniques.
The big point from last time
If we add two equations together, the result is an equation than contains the common s
MA 3280 Lecture 04 - Gaussian Elimination
Wednesday, January 29, 2014.
Objectives: Solution techniques for larger systems.
Solving larger systems of linear equations with the techniques discussed on Monday (which is often called
Gaussian elimination ) is
MA 3280 Lecture 06 - Rank and Row Equivalence
Monday, February 3, 2014.
Objectives: Summary of echelon matrices, row canonical form, row equivalence from Section 3.7 starting on
page 70, and introduce matrix multiplication.
Note: A matrix may represent a
MA 3280 Lecture 14 - Spanning Sets
Friday, February 28, 2014.
Objectives: Material from 4.1: Introduction to vector spaces.
Spanning sets
Example 1. Consider the vector space P2 (t). Note that all the vectors in P2 (t) can be expressed as a linear
combina