12:58
What's the big idea of Linear Algebra? **Course Intro**
Dr. Trefor Bazett
5:28
What is a Solution to a Linear System? **Intro**
8:19
Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically
3:10
Rewriting a Linear System using Matrix Notation
7:27
Using Elementary Row Operations to Solve Systems of Linear Equations
9:35
Using Elementary Row Operations to simplify a linear system
6:30
Examples with 0, 1, and infinitely many solutions to linear systems
6:41
Row Echelon Form and Reduced Row Echelon Form
11:05
Back Substitution with infinitely many solutions
13:37
The Gaussian Algorithm Visualized
6:40
What is a vector? Visualizing Vector Addition & Scalar Multiplication
9:34
Introducing Linear Combinations & Span
5:00
How to determine if one vector is in the span of other vectors?
6:59
Matrix-Vector Multiplication and the equation Ax=b
4:33
Matrix-Vector Multiplication Example
8:25
Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac
14:01
The Big Theorem, Part I
4:48
Writing solutions to Ax=b in vector form
6:21
Geometric View on Solutions to Ax=b and Ax=0.
7:53
Three nice properties of homogeneous systems of linear equations
8:16
Linear Dependence and Independence - Geometrically
6:39
Determining Linear Independence vs Linear Dependence
7:10
Making a Math Concept Map | Ex: Linear Independence
5:16
Transformations and Matrix Transformations
Three examples of Matrix Transformations
8:28
Linear Transformations
3:04
Are Matrix Transformations and Linear Transformation the same? Part I
6:37
Every vector is a linear combination of the same n simple vectors!
8:49
Matrix Transformations are the same thing as Linear Transformations
9:18
Finding the Matrix of a Linear Transformation
9:30
One-to-one, Onto, and the Big Theorem Part II
10:12
The motivation and definition of Matrix Multiplication
5:37
Computing matrix multiplication
14:00
Visualizing Composition of Linear Transformations **aka Matrix Multiplication**
7:20
Elementary Matrices
7:12
You can "invert" matrices to solve equations...sometimes!
Finding inverses to 2x2 matrices is easy!
Find the Inverse of a Matrix
8:45
When does a matrix fail to be invertible? Also more "Big Theorem".
8:12
Visualizing Invertible Transformations (plus why we need one-to-one)
6:38
Invertible Matrices correspond with Invertible Transformations **proof**
9:14
Determinants - a "quick" computation to tell if a matrix is invertible
3:43
Determinants can be computed along any row or column - choose the easiest!
8:11
Vector Spaces | Definition & Examples
12:50
The Vector Space of Polynomials: Span, Linear Independence, and Basis
6:26
Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples
5:03
The Span is a Subspace | Proof + Visualization
10:41
The Null Space & Column Space of a Matrix | Algebraically & Geometrically
3:53
The Basis of a Subspace
9:45
Finding a Basis for the Nullspace or Column space of a matrix A
2:37
Finding a basis for Col(A) when A is not in REF form.
6:34
Coordinate Systems From Non-Standard Bases | Definitions + Visualization
3:52
Writing Vectors in a New Coordinate System **Example**
What Exactly are Grid Lines in Coordinate Systems?
5:11
The Dimension of a Subspace | Definition + First Examples
3:27
Computing Dimension of Null Space & Column Space
4:02
The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
7:55
Changing Between Two Bases | Derivation + Example
7:42
Visualizing Change Of Basis Dynamically
8:55
Example: Writing a vector in a new basis
9:09
What eigenvalues and eigenvectors mean geometrically
4:36
Using determinants to compute eigenvalues & eigenvectors
7:33
Example: Computing Eigenvalues and Eigenvectors
9:47
A range of possibilities for eigenvalues and eigenvectors
6:28
Diagonal Matrices are Freaking Awesome
7:30
How the Diagonalization Process Works
3:01
Compute large powers of a matrix via diagonalization
10:08
Full Example: Diagonalizing a Matrix
14:10
COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
9:46
Visualizing Diagonalization & Eigenbases
8:47
Similar matrices have similar properties
9:59
The Similarity Relationship Represents a Change of Basis
7:23
Dot Products and Length
8:14
Distance, Angles, Orthogonality and Pythagoras for vectors
4:50
Orthogonal bases are easy to work with!
3:57
Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement
11:07
The geometric view on orthogonal projections
11:25
Orthogonal Decomposition Theorem Part II
7:05
Proving that orthogonal projections are a form of minimization
11:21
Using Gram-Schmidt to orthogonalize a basis
6:18
Full example: using Gram-Schmidt
7:28
Least Squares Approximations
5:36
Reducing the Least Squares Approximation to solving a system
