Math 308B - Matrix Algebra

Lecture Summaries

10F - 06/02/17

• Review for the Final Exam
• The Final Exam is Wednesday, June 7 at 8:30am in Architecture 147
• Remember to bring your Photo ID

10W - 05/31/17

• Triangular Change of Basis
• Least Squares Regression: Minimize Distance to a Subspace
• Linear Regression: Finding the Best Fit Line
• Orthogonality and the Normal Equations
• Quadratic Regression: The Best Fit Parabola
• The Final Exam is Wednesday, June 7 at 8:30am in Architecture 147

9F - 05/26/17

• Projecting a Vector onto a Subspace: Orthogonal Bases
• Orthonormal Bases
• Computing an Orthoganal Basis: The Gram-Schmidt Process

9W - 05/24/17

• Dot Product: Orthogonal Complement of a Subspace
• The Orthogonal Complement is a Subspace
• Calculating the Orthogonal Complement
• Orthogonal Sets of Vectors
• Orthogonal Sets are Linearly Independent
• Projecting a Vector onto another Vector
• Projecting a Vector onto a Subspace: Orthogonal Bases

9M - 05/22/17

• Change of Basis in a Subspace
• The Dot Product
• Magnitude and Distance
• Orthogonality and the Pythagorean Theorem

8F - 05/19/17

• Comments on the Midterm: Rank and Column Space
• Change of Basis: The Standard Basis
• The Change of Basis Matrix
• Using Cofactors to Invert a Matrix
• Changing Between Two Non-standard Bases
• Example: Diagonalization of a Matrix

8W - 05/17/17

• Midterm 2
• Here are the 10th and 11th Discussion Problems

8M - 05/15/17

• Midterm review: the exam covers 3.2,3.3,4.1-4.3,5.1,6.1

7F - 05/12/17

• Information about the Midterm: It will cover through Eigenvectors
• More on Eigenvectors and Eigenvalues
• Computing Eigenvalues: Using Row Operations to Simplify Calculations
• Finding Integer Roots of Polynomials

• Determinants and the Transpose
• Minors and Cofactors
• Fixed Points of a Linear Transformation
• Eigenvalues: the Characteristic Polynomial
• Eigenvectors: Calculating λ-Eigenspaces
• Complex Eigenvalues
• Algebraic Multiplicity and Geometric Multiplicity

7M - 05/08/17

• Calculating the Determinant of a 3x3 Matrix: Expanding along the Second Column
• Expanding along any Row or Column: Signs and Minors
• Linear Forms on n-space: The Dot Product
• Alternating Forms: The Real Definition of the Determinant
• Determinants and Elementary Row Operations
• Determinant of a Triangular Matrix: det(I)=1
• Determinants and Nonsingularity
• Determinants of Products and Inverses

6F - 05/05/17

• Dimension of a Subspace: Showing two Subspaces are Equal
• Nullity of a Matrix
• Row Space and Column Space have the same Dimension
• Rank of a Matrix
• Rank plus Nullity equals the dimension of the Domain
• Introduction to Determinants: The 2x2 Case
• Calculating the Determinant of a 3x3 Matrix: Expanding along the First Row

6W - 05/03/17

• The Standard Basis of Euclidean n-space
• Unique Representation of a Vector in terms of a Basis
• Dimension of a Subspace
• Proving that every Basis has the same size
• Expanding an Independent Set to a Basis
• Here are the 8th and 9th Discussion Problems

6M - 05/01/17

• The Basis of a Subspace
• Computing a Basis for a Null Space
• Row Space and Column Space of a Matrix
• Using Row Reduction to Compute a Basis from a Spanning Set
• Deleting Vectors from a Spanning Set to get a Basis
• Dimension of a Subspace

5F - 04/28/17

• Subspaces: Definition
• Checking if a Subset is a Subspace: The Zero Vector
• Checking Closure under Scalar Multiplication
• Checking Closure under Addition
• Using Geometry to Visualize Subspaces
• Null Space of a Matrix and Kernel of a Linear Transformation
• Range of a Linear Transformation: Spans are Subspaces

5W - 04/26/17

• Calculating the Matrix Inverse
• The Inverse of a 2x2 Matrix: The Determinant
• Singular and Nonsingular Matrices: The Big Theorem
• Solving a Linear System using the Inverse Matrix
• Properties of the Matrix Inverse: More Basic Proof Techniques
• Here are the 6th and 7th Discussion Problems

5M - 04/24/17

• The Transpose of a Matrix: Properties
• Powers of a Matrix
• Diagonal and Triangular Matrices
• Basic Proof Techniques: Mathematical Induction
• Inverse Matrices and Linear Transformations
• Matrix Inverse: Computations
• Calculating the Matrix Inverse using Gauss-Jordan Elimination

4F - 04/21/17

• Matrix Algebra: Vector Space Properties
• Matrix Multiplication: Composition of Linear Transformations
• Properties of Matrix Multiplication: Possible Difficulties
• The Multiplicative Identity and Diagonal Matrices
• The Transpose of a Matrix

4W - 04/19/17

• Midterm 1
• Here are the 4th and 5th Discussion Problems

4M - 04/17/17

• Midterm review: the exam covers 1.1,1.2,1.4-3.1

3F - 04/14/17

• Example: Not all Lines are Linear
• Linear Transformations: One-to-one and Onto Functions
• One-to-one Linear Transformations and Homogeneous Linear Systems
• Onto Linear Transformations and Spanning Sets of Vectors
• More Parts to the Big Theorem: Kernel of a Linear Transformation
• Linear Transformations and Geometry: Lines go to Lines

3W - 04/12/17

• Linear Transformations: Definition and Examples
• Domain and Codomain
• Matrix Multiplication gives a Linear Transformation
• How to get the Matrix for a Linear Transformation
• The Range of a Linear Transformation: Column Span of a Matrix
• Checking Linearity
• Onto Functions

3M - 04/10/17

• More on Linear Independence: an Example
• A little Matrix Algebra: Linearity
• General Solutions and the Associated Homogeneous Linear System
• Many Short Theorems in Chapter 2.3: Some Proofs
• The "Big Theorem"

2F - 04/07/17

• Information about Midterm 1
• Can 2 vectors span 3-space?
• Linear Dependence and Independence
• Is a set of vectors linearly independent?
• Many short theorems in Chapter 2.3.
• A little matrix algebra

2W - 04/05/17

• More on Linear Combinations
• Matrix Equation of a Linear System
• The Span of a Set of Vectors
• Is a given vector in a span?
• Does a set of vectors span n-space?
• How many vectors does it take to span n-space?
• Statement of Theorem 2.7.
• Here are the 2nd and 3rd Discussion Problems

2M - 04/03/17

• Vectors and Euclidean Space, Linear Combinations
• Linear Systems and Vector Equations
• Writing the General Solution as a Linear Combination
• Is a Vector a Linear Combination of some Given Vectors?
• An application from Operations Research

1F - 03/31/17

• Application: Balancing Chemical Equations
• Application: Traffic Flow
• Choosing a "good" particular solution

1W - 03/29/17

• More on Gaussian Elimination and Gauss-Jordan Elimination
• Leading and Independent Variables
• Inconsistent Linear Systems
• Proof of Theorem 1.2
• Homogeneous Linear Systems
• Here is the 1st Discussion Problem

1Tu - 03/28/17

• Introduction to Linear Algebra.
• Augmented Matrix and Elementary Row Operations.
• Gaussian Elimination and Row Echelon Form.
• Gauss-Jordan Elimination and Reduced Row Echelon Form.

• Overview of course
• Introduction to Linear Systems.
• You need WebAssign for this class. Here is a link with more information.
• (This link is for Math 124, but it should be helpful if you are just getting started with WebAssign.)