Episodes and Slides:

Notes:
  • Unless things go horribly wrong, all lectures will be videoed and posted here.
  • Binge-watching the course will be possible via the course playlist on youtube. Please watch and think responsibly.
  • There will be an occasional set of lecture slides appearing in the course. They'll be available on this page at the end and crosslinked with the relevant episode.

Episode index:

S8E01: Overview, motivation, and the Row, Column, and Matrix Picture.
Three ways of looking at Ax=b.

S8E02: Gaussian Elimination the one True Way.
Our matrix A connects with its inner matrices, U and R.

S8E03: Enter the Elimination Matrix.
We make matrices do the work for us.

S8E04: Advanced matrix wrangling and a start on inverses.
The manifold ways of multiplying matrices.

S8E05: Our quest is the inverse of A.
We explain how the Gauss-Jordan method works using elimination, permutation, and pivot matrices.

S8E06: Triangle x Triangle = Rectangle.
We factorize matrices into a product of lower and upper triangular matrices.

S8E07: More LU Truth, then the surprises of transposes and symmetry.
We explore the simple but surprisingly powerful and important transpose operation.

S8E08: Danger Will Robinson: We are entering Column Space!
We look at vector spaces and subspaces, and we then uncover the first of the Big Picture's four fundamental and joyous subspaces, Column Space. Don't worry, we won't get lost.

S8E09: Nullspace versus Buffy the Vector Slayer.
We uncover Nullspace, our second fundamental subspace, find out about pivot and free variables, and define matrix rank.

S8E10: Subspacejam
More on how to find column space and nullspace, how these subspaces connect to the full solution of Ax=b, and a taxonomy of reduced row echelon forms.

S8E11: All your bases are belong to us.
Dimensions and the Bigness of Subspaces.

S8E12: It came from Row Space!
We enter a new land.

S8E13: The Unbuildable Reality of Left Nullspace.
How A and AT give you all the joy in the world.

S8E14: Finding the Big Picture of Ax=b for three example matrices.
All the subspaces you've come to know and love plus a recap of the four types of A.

S8E15: The Amazing Normal Equation.
Approximating Ax=b when no solution exists through the art of projection.

S8E16: Making orthogonal bases.
The Gram-Schmidt Process and a new factorization A=QR.

S8E17: The Magic of Eigenthings.
A start on all things Eigen with some motivation through powers of square matrices and their use in understanding probabilistic processes (distracted texter on a networks), differential equations, and difference equations.

S8E18: The Glorious Eigenvalue Equation and a start on Determinants.
We construct and solve the eigenvalue equation, find the nullspace equation yet again. Wedecide we need something better for finding out when a matrix has no inverse, and this opens a big box with the words "Caution: Matrix Determinants" on the side.

S8E19a: The peculiar properties of determinants.
Starting from three assertions about determinants, we show how standard row operations can be used to find many useful results.

S8E19b: Computing determinants using the Way of the Cofactor.
We go through a method for systematically computing determinants that is especially useful for finding eigenvalues.

S8E19c: Cramer's rule.
The surprising Cramer's rule gives us a formula solution for Ax=b and a formula for A-1. Cofactors appear again. Computing determinants is computationally wearying so the importance here is more theoretical than practical.

S8E20: Diagonalization is just the best.
We warm up with a sampling of sneaky tricks about eigengthings, and then we jump into the glorious truth of matrix diagonalization.

S8E21: Symmetric matrices love diagonalization.
Finding all the Fibonacci numbers, the golden ratio, how change of basis works, and the incredible properties of symmetric matrices.

S8E22: The strange connections between pivots and eigenvalues.
Also: Why symmetric matrices have orthogonal eigenvectors, a charming result for traces, and a start on positive definite matrices.

S8E23: Positively, Definitely Matrices.
More surprises: Row reduction = Completing the Square, how matrices can help with surfaces and curves and functions, the Principal Axis Theorem.

S8E24: Singular Value Decomposition is Awesome.
We explain how SVD is possible for all matrices, using much matrix-fu, and we tackle one feisty two by two.

S8E25: How Singular Value Decomposition is all about Scotland.
Example calculations, spheres and ellipsoids, SVD and network search, SVD and image compression, SVD and the Scottish Fabric Image Guessing Game (1:05:00).


Episodes:

Episode 01: Overview, motivation, and the Row, Column, and Matrix Picture.
Three ways of looking at Ax=b.
Duration: 1:15:37


Slides covered:
01. Introduction to Matrixology.

Episode 02: Gaussian Elimination the one True Way.
Our matrix A connects with its inner matrices, U and R.
Duration: 1:16:17




Episode 03: Enter the Elimination Matrix.
We make matrices do the work for us.
Duration: 1:15:49



Episode 04: Advanced matrix wrangling and a start on inverses.
The manifold ways of multiplying matrices.
Duration: 1:15:04




Episode 05: Our quest is the inverse of A.
We explain how the Gauss-Jordan method works using elimination, permutation, and pivot matrices.
Duration: 1:14:24



Episode 06: Triangle x Triangle = Rectangle.
We factorize matrices into a product of lower and upper triangular matrices.
Duration: 1:10:50


And we overcome some recording pathologies.


Episode 07: More LU Truth, then the surprises of transposes and symmetry.
We explore the simple but surprisingly powerful and important transpose operation.
Duration: 1:13:44


Slides covered:
02. Review for Challenge Level 1.

Episode 08: Danger Will Robinson: We are entering Column Space!
We look at vector spaces and subspaces, and we then uncover the first of the Big Picture's four fundamental and joyous subspaces, Column Space. Don't worry, we won't get lost.
Duration: 1:15:16


We also talk about happiness.


Episode 09: Nullspace versus Buffy the Vector Slayer.
We uncover Nullspace, our second fundamental subspace, find out about pivot and free variables, and define matrix rank.
Duration: 1:17:22



Episode 10: Subspacejam
More on how to find column space and nullspace, how these subspaces connect to the full solution of Ax=b, and a taxonomy of reduced row echelon forms.
Duration: 1:14:03




Episode 11: All your bases are belong to us.
Dimensions and the Bigness of Subspaces.
Duration: 1:14:26


Also: the emotional trajectories of movies.

Episode 12: It came from Row Space!
We enter a new land.
Duration: 1:14:31




Episode 13: The Unbuildable Reality of Left Nullspace.
How A and AT give you all the joy in the world.
Duration: 1:16:52



Episode 14: Finding the Big Picture of Ax=b for three example matrices.
All the subspaces you've come to know and love plus a recap of the four types of A.
Duration: 1:21:40


Scanned notes are here.


Episode 15: The Amazing Normal Equation.
Approximating Ax=b when no solution exists through the art of projection.
Duration: 1:12:14


Various recording disasters made this a more complicated production.
Related Webisodes:
1.The Normal Equation for the example of brave Westley. (7:46)
2. Let's project a vector onto a subspace. (13:04)
3. When is ATA invertible? (7:03)

Episode 16: Making orthogonal bases.
The Gram-Schmidt Process and a new factorization A=QR.
Duration: 1:18:23


Related Webisodes:
1.Orthogonal Matrices. (6:36)
2. Solving Ax=b with A=QR. (7:37)

Episode 17: The Magic of Eigenthings.
A start on all things Eigen with some motivation through powers of square matrices and their use in understanding probabilistic processes (distracted texter on a networks), differential equations, and difference equations.
Duration: 1:13:00



Episode 18: The Glorious Eigenvalue Equation and a start on Determinants.
We construct and solve the eigenvalue equation, find the nullspace equation yet again. Wedecide we need something better for finding out when a matrix has no inverse, and this opens a big box with the words "Caution: Matrix Determinants" on the side.
Duration: 1:18:46


Related Webisode:
1. Determinants from the ground up. (19:23) [download notes]
Complete set of notes on determinants is available here.


Episode 19a: The peculiar properties of determinants.
Starting from three assertions about determinants, we show how standard row operations can be used to find many useful results.
Duration: 41:09


Notes are here.

Episode 19b: Computing determinants using the Way of the Cofactor.
We go through a method for systematically computing determinants that is especially useful for finding eigenvalues.
Duration: 25:50


Notes are here.


Episode 19c: Cramer's rule.
The surprising Cramer's rule gives us a formula solution for Ax=b and a formula for A-1. Cofactors appear again. Computing determinants is computationally wearying so the importance here is more theoretical than practical.
Duration: 17:03


Notes are here.

Episode 20: Diagonalization is just the best.
We warm up with a sampling of sneaky tricks about eigengthings, and then we jump into the glorious truth of matrix diagonalization.
Duration: 1:13:45




Episode 21: Symmetric matrices love diagonalization.
Finding all the Fibonacci numbers, the golden ratio, how change of basis works, and the incredible properties of symmetric matrices.
Duration: 1:15:46



Episode 22: The strange connections between pivots and eigenvalues.
Also: Why symmetric matrices have orthogonal eigenvectors, a charming result for traces, and a start on positive definite matrices.
Duration: 1:10:06


Messed up the sound recording. It's serviceable.


Episode 23: Positively, Definitely Matrices.
More surprises: Row reduction = Completing the Square, how matrices can help with surfaces and curves and functions, the Principal Axis Theorem.
Duration: 1:12:22



Episode 24: Singular Value Decomposition is Awesome.
We explain how SVD is possible for all matrices, using much matrix-fu, and we tackle one feisty two by two.
Duration: 1:16:09


Some issues with tracking; overlaid a few blackboard photos (boardshots).


Episode 25: How Singular Value Decomposition is all about Scotland.
Example calculations, spheres and ellipsoids, SVD and network search, SVD and image compression, SVD and the Scottish Fabric Image Guessing Game (1:05:00).
Duration: 1:15:54


Slow clap for Linear Algebra at 1:00:00 courtesy @chrisdanforth.


Slides:

Notes:
  • The slides are clickable pdf's, with links to relevant web pages, section links in the sidebar, and navigation icons at the bottom of each slide. The reference section for each set of slides includes links to papers, and superscript citation numbers link to the reference section if present (please let me know if any links behave badly).
  • The `lecture' pdf's include all the incremental reveals whereas the `slides' pdf's have flattened frames and are better for reading online.
  • If you want a printed copy, the handouts provide condensed and collapsed versions of the slides.

01. Introduction to Matrixology.
[slides]     [handout]     [lecture]
       
Covered in this episode:
S8E01: Overview, motivation, and the Row, Column, and Matrix Picture.

02. Review for Challenge Level 1.
[slides]     [handout]     [lecture]
       
Covered in this episode:
S8E07: More LU Truth, then the surprises of transposes and symmetry.


03. Singular Value Decomposition.
[slides]     [handout]     [lecture]
       
Covered in this episode:
S8E25: How Singular Value Decomposition is all about Scotland.


(Note: I use the splendid LaTeX class beamer coupled with some surreptitious perl scripts to generate these slides.)

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