| Unit |
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 row-echelon form (RREF),
augmented matrix in RREF,
Gauss-Jordan 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 right-hand sides,
worked examples
|
| 6 |
Inverse Matrix and Matrix Algebra |
left and right inverses,
finding inverse matrices,
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,
rank-nullity 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,
change-of-basis matrix,
data analysis,
differentiation,
worked examples
|
| 12 |
Applications |
low-rank matrix approximation,
singular value decomposition(SVD),
least squares approximation,
facial recognition
|