  # Linear Algebra I: From Linear Equations to Eigenspaces

### Course Description

This course contains 47 short video lectures by Dr. Bob on basic and advanced concepts from Linear Algebra. He walks you through basic ideas such as how to solve systems of linear equations using row echelon form, row reduction, Gaussian-Jordan elimination, and solving systems of 2 or more equations using determinants, Cramer's rule, and more.

He also looks over concepts of vector spaces such as span, linear maps, linear combinations, linear transformations, basis of a vector, null space, changes of basis, as well as finding eigenvalues and eigenvectors.

Finally, he finishes the course covering some advanced concepts involving eigenvectors, including the diagonalization of the matrix, the power formula for a matrix, solving Fibonacci numbers using linear algebra, inner product on R^n, orthogonal transformations, Gram-Schmidt orthogonalization, QR-decomposition, the spectral theorem, and much more. Dr. Bob teaches Cramer's rule for solving systems of three linear equations.
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### Video Lectures & Study Materials

# Lecture Play Lecture
1 Row Reduction for a System of Two Linear Equations (5:06) Play Video
2 Solving a 2x2 SLE Using a Matrix Inverse (3:44) Play Video
3 Solving a SLE in 3 Variables with Row Operations 1 (8:05) Play Video
4 Solving a SLE in 3 Variables with Row Operations 2 (8:24) Play Video
5 Consistency of a System of Linear Equations (3:11) Play Video
6 Inverse of 3 x 3 Matrix Using Row Operations 1 (5:58) Play Video
7 Inverse of 3x3 Matrix Using Row Operations 2 (6:25) Play Video
8 Inverse of 4x4 Matrix Using Row Operations (3:41) Play Video
9 Example of Determinant Using Row Echelon Form (6:56) Play Video
10 Inverse of 3 x 3 Matrix Using Adjugate Formula (3:20) Play Video
11 Inverse of 4x4 Matrix Using Adjugate Formula (4:18) Play Video
12 Cramer's Rule for Three Linear Equations (4:50) Play Video
13 Determinant of a 4 x 4 Matrix Using Cofactors (4:24) Play Video
14 Determinant of a 4 x 4 Matrix Using Row Operations (4:23) Play Video
15 Examples of Linear Maps (6:46) Play Video
16 Example of Linear Combination (4:19) Play Video
17 Example of Linear Combination (Visual) (4:08) Play Video
18 Evaluating Linear Transformations Using a Basis (3:28) Play Video
19 Linear Transformations on R^2 (11:38) Play Video
20 Example of Checking for Basis Property (4:33) Play Video
21 Example of Basis for a Null Space (4:35) Play Video
22 Example of Basis for a Span (6:37) Play Video
23 Example of Linear Independence Using Determinant (3:05) Play Video
24 Example of Kernel and Range of Linear Transformation (6:31) Play Video
25 Linear Transformations: One-One (9:00) Play Video
26 Linear Transformations: Onto (8:25) Play Video
27 Example of Change of Basis (7:11) Play Video
28 Eigenvalues and Eigenvectors (9:45) Play Video
29 Example of Eigenvector: Markov Chain (13:26) Play Video
30 Example of Diagonalizing a 2 x 2 Matrix (7:03) Play Video
31 Example of Power Formula for a Matrix (7:11) Play Video
32 The Fibonacci Numbers Using Linear Algebra (HD Version) (12:25) Play Video
33 Vector Length in R^n (12:37) Play Video
34 The Standard Inner Product on R^n (10:49) Play Video
35 Example of Fourier's Trick (6:06) Play Video
36 Example of Orthogonal Complement (4:21) Play Video
37 Orthogonal Transformations 1: 2x2 Case (14:22) Play Video
38 Orthogonal Transformations 2: 3x3 Case (15:14) Play Video
39 Example of Gram-Schmidt Orthogonalization (9:52) Play Video
40 QR-Decomposition for a 2x2 Matrix (10:33) Play Video
41 Beyond Eigenspaces: Real Invariant Planes (16:12) Play Video
42 Beyond Eigenspaces 2: Complex Form (14:22) Play Video
43 Spectral Theorem for Real Matrices: General 2x2 Case (14:14) Play Video
44 Spectral Theorem for Real Matrices: General nxn Case (15:34) Play Video
45 Example of Spectral Theorem (3x3 Symmetric Matrix) (7:54) Play Video
46 Example of Spectral Decomposition (10:13) Play Video
47 Example of Diagonalizing a Symmetric Matrix (Spectral Theorem) (10:06) Play Video