Linear Algebra for Scientific Thinkers

Week Topic Parts

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 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, 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

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