# Math 235 Class Prep Videos

On this page, you will find links to class prep videos, which are on YouTube.

These videos should be watched before attending class.

Section Video
1.1 a) Systems of Equations (5:18)
b) Geometry of Linear Systems (8:39)
c) Augmented Matrices of Linear Systems (5:01)
1.2 a) Solving Systems using Matrices (12:24)
b) Reducing Rob's Favorite Matrix to Echelon Form (16:49)
c) Solving the System Associated to Rob's Favorite Matrix (7:36)
d) Infinitely Many Solutions and Free Variables (13:58)
e) A Remark on Consistency (4:46)
1.3
 Required Video: Vectors, Linear Combinations, and Spans (28:37) Recommended Videos: a) Building Intuition: Linear Combinations and Chess I (8:30) b) Building Intuition: Linear Combinations and Chess II (14:57) c) Building Intuition: Combos and Carbs (14:47)
1.4
 Required Video: Multiplication and Matrix Equations (17:35) Recommended Videos: a) Multiplication: Motivation and Definition (8:41) b) Multiplying Matrices and Vectors: Row-Vector Rule (11:38) c) Matrix Equations (13:02)
1.5
 Required Video: Homogeneous Systems (16:49) Recommended Videos: a) Example: Ants and Anti-Ants I (7:04) b) Example: Ants and Anti-Ants II (11:01)
1.7 Linear Independence (24:23)
1.8 Linear Transformations (18:22)
1.9
 Required Videos: a) The Standard Matrix of a Linear Transformation (14:32) b) 90° Counterclockwise Rotation of R2 (7:27) c) One-to-one and Onto Transformations (17:28) Recommended Videos: a) Using Rotations of R2 to Prove Trig Identities (6:10) b) Rotations of R3 (7:05)
2.1 Matrix Operations (18:10)
2.2
 Required Video: Matrix Inverses (22:46) Recommended Video: Elementary Matrices and the Proof of the Matrix Inverse Algorithm (12:58)
2.3 a) Invertible Linear Transformations (9:53)
b) Invertible Matrix Theorem (19:07)
Chapters 1 and 2 Supplemental Videos
 Some Review: a) Review of Equivalent Statements (20:44) b) Dictionary Between Linear Transformations and Properties of Vectors I (11:01) c) Dictionary Between Linear Transformations and Properties of Vectors II (16:40) Applications: a) RBG and Composing Linear Transformations (12:41) b) Using Matrices to Understand How Cube Corner Reflectors (Bike Reflectors) Work (11:55) c) An Experiment: Demonstrating How Cube Corner Reflectors Work (1:29) d) Row Reducing a Baklava Matrix (1:18) For the Curious: a) Right and Left Inverses I (10:56) b) Right and Left Inverses II (8:00) c) Symmetry Challenge Problems (18:09)
3.1 a) Intro to Determinants (19:08)
b) Cofactor Expansions and Triangular Matrices (14:37)
3.2 a) The Effect of Row Operations on Determinants (6:48)
b) Determinant of Invertible Matrices and Properties of Determinants (11:16)
3.3 a) Cramer's Rule (15:47)
b) The Adjugate and a formula for the Inverse of a Matrix (14:48)
c) Interpreting Determinants as Areas and Volumes (21:53)
d) Linear Maps and Determinants (6:19)
Chapter 3 Supplemental Videos a) Permutation Matrices and Determinants I (16:19)
b) Permutation Matrices and Determinants II (18:23)
c) Permutation Matrices and Determinants III (7:59)
4.1 Introduction to Vector Spaces (14:45)
4.2 Null Spaces, Column Spaces, and Linear Transformations (9:26)
4.3 Linear Independence and Bases (14:33)
4.4 Coordinate Systems (10:26)
4.5 Dimension of Vector Spaces (11:59)
4.6 Row Space and Rank (9:17)
Chapter 4 Supplemental Videos a) Standard Matrices for Linear Transformations between General Vector Spaces I (10:04)
b) Standard Matrices for Linear Transformations between General Vector Spaces II (an example) (15:46)
c) The Vector Space Hom(V,W) (13:47)
5.1 a) Intro to Eigenvalues and Eigenvectors (16:07)
b) Eigenspaces (9:07)
5.2 a) The Characteristic Equation (10:12)
b) Eigenvalues and Determinants (10:28)
5.3 Diagonalization (21:36)
5.5 Complex Eigenvalues (8:40)
Chapter 5 Supplemental Videos a) Markov Chains I (23:11)
b) Markov Chains II (15:24)
6.1
 Required Video: Inner Product (Dot Product) on Rn and Orthogonal Vectors (13:10) Recommended Videos: a) Inner Products on General Vector Spaces I (14:43) b) Inner Products on General Vector Spaces II (4:43) c) Inner Products on General Vector Spaces III (10:35)
6.2 Orthogonal Projections (22:14)
6.3 Orthogonal Decomposition Theorem (13:52)
6.4 a) The Gram-Schmidt Process (18:46)
b) QR Factorizations (12:47)

# Supplemental Videos

Below are supplemental videos created by Navid Mirzaei, which were used during the summer semester.

These videos are purely supplemental and do not need to be watched before coming to class.