The Matrix

* What is a matrix?  Traditional answer

** List of row-lists, list of column-lists

*** List of row-lists, list of column-lists (Quizzes)

** The matrix revealed

*** Rows, columns, and entries

*** Dict-of-rows/dict-of-columns representations

** Our Python implementation

*** Identity matrix

*** Converting between representations

*** Module matutil

*** The Mat class

**** Using Mat

**** Assertions in Mat

**** Testing Mat with doctests

* Column space and row space

* Transpose

** Transpose (and Quiz)

* Matrices as vectors

* Matrix-vector and vector-matrix multiplication in terms of linear combinations

** Matrix-vector multiplication in terms of linear combinations

*** Matrix-vector multiplication in terms of linear combinations: Lights Out

*** Solving a matrix-vector equation

*** Solving a matrix-vector equation: Lights Out

*** Solving a matrix-vector equation: 2 × 2 special case

*** The solver module

** Vector-matrix multiplication in terms of linear combinations

*** Vector-matrix multiplication in terms of linear combinations: JunkCo

** Solving a matrix-vector equation

*** The solver module, and floating-point arithmetic

*** Checking the output from solve(A, b)

* Matrix-vector multiplication in terms of dot-products

** Applications of dot-product definition of matrix-vector multiplication

** Formulating a system of linear equations as a matrix-vector equation

*** Matrix-vector equation for sensor node

** Triangular matrix

** Computing sparse matrix-vector product

* Matrix-matrix multiplication

** Matrix-matrix multiplication: vector-matrix definition

*** Matrix-matrix multiplication: vector-matrix interpretation

** Matrix-matrix multiplication: matrix-vector definition

** Matrix-matrix multiplication: Dot-product definition

** Matrix-matrix multiplication: transpose

** Matrix-matrix multiplication: Column vectors

** Matrix-matrix multiplication: Row vectors

** Algebraic properties of matrix-vector multiplication

* Null space of a matrix

** Solution space of a matrix-vector equation

** Error-correcting codes

*** Error-correcting codes: Block codes

*** Error-correcting codes: Linear binary block codes

*** Hamming Code

* Matrices and their functions

** From matrix to function

 ** From function to matrix

*** From function to matrix: horizontal scaling

*** From function to matrix: rotation by 90 degrees

*** From function to matrix: rotation by θ degrees

*** From function to matrix: translation

*** From function to matrix: identity function

*** Diagonal matrices

* Linear functions: Which functions can be expressed as a matrix-vector product?

** Which functions are linear?

*** A linear function maps zero vector to zero vector

*** Linear functions: Pushing linear combinations through the function

** From function to matrix, revisited (proof algorithm works)

** Linear functions and zero vectors: Kernel

** Kernel and one-to-one

** Linear functions that are onto?

** The image of a linear function is a vector space

** Linear functions that are onto?

* Inner product / Outer product of matrices

* Matrix-matrix multiplication and function composition

** Associativity of matrix-matrix multiplication

* From function inverse to matrix inverse

** Matrix inverse

** Invertible matrices: why care?

** Product of invertible matrices is an invertible matrix

** Matrix inverse

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