Week 
Topic 
Parts 
1 
Introduction and Basic Concepts 
digital images,
modelling Lights Out,
tuple arithmetic,
set notation,
functions,
additive and multiplicative inverses,
fields,
complex numbers,
the complex plane,
Euler's identity,
worked examples

2 
Complex Numbers Arithmetic 
addition and multiplication,
subtraction and division,
complex conjugate,
modulus of a complex number,
polar form,
conversion between forms,
de Moivre's Theorem,
finding nth roots,
quadratic equations,
why complex numbers,
worked examples

3 
Systems of Linear Equations 
setting up a system for Lights Out,
linear equations introduction,
simple systems,
method of substitution,
elementary operations,
matrix representations,
column view,
worked examples

4 
Row Reduction 
augmented matrix,
reduced rowechelon form (RREF),
augmented matrix in RREF,
GaussJordan elimination,
describing solution sets,
homogeneous systems,
solving 3 x 3 Lights Out,
worked examples

5 
Matrix Multiplication 
linear transformation view,
matrix multiplication,
associativity of matrix multiplication,
identity matrix,
row reduction as matrix multiplication,
elementary matrices example,
multiple righthand sides,
worked examples

6 
Inverse Matrix and Matrix Algebra 
left and right inverses,
finding matrix inverses,
inverse of a product,
generating invertible matrices,
singular matrix,
matrix properties,
transpose of a matrix,
matrix powers,
image manipulation,
2D graphics,
worked examples

7 
Determinants 
permutations,
inversions,
definition,
special matrices,
determinant of a product,
properties,
computing via row reduction,
cofactor expansion,
Cramer's rule,
worked examples

8 
Vector Spaces 
motivation,
definition,
examples and subspaces,
linear combination and span,
infinite dimension example,
different sets spanning the same set,
visualizing \(\mathbb{R}^2\),
dot product,
worked examples

9 
Basis and Dimension 
linear independence,
basis and dimension,
dimensions of subspaces,
basis for nullspace,
column space and row space,
ranknullity theorem,
tuple representation,
orthonormal bases,
Lights Out solution count,
worked examples

10 
Eigenvalues and Eigenvectors 
matrix powers magic,
eigenvalues and eigenvectors,
algebraic and geometric multiplicities,
diagonalization,
recurrence relation,
symmetric matrices,
orthogonal diagonalization example,
worked examples

11 
Linear Transformations 
definition,
kernel,
surjection, injection, bijection,
invertible linear transformations,
matrix representation,
changeofbasis matrix,
data analysis,
differentiation,
worked examples

12 
Applications 
lowrank matrix approximation,
singular value decomposition(SVD),
least squares approximation,
facial recognition
