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