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
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
7W - 05/10/17
- 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
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.
1M - 03/27/17
- 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.)