Introductions
  • What is linear algebra?
  • Linear algebra applications
  • An enticing start to a linear algebra course!
  • How best to learn from this course
  • Maximizing your Udemy experience
  • Using MATLAB, Octave, or Python in this course
Vectors
  • Exercises + code
  • Algebraic and geometric interpretations of vectors
  • Vector addition and subtraction
  • Vector-scalar multiplication
  • Vector-vector multiplication: the dot product
  • Dot product properties: associative, distributive, commutative
  • Code challenge: dot products with matrix columns
  • Vector length
  • Vector length in MATLAB
  • Vector length in Python
  • Dot product geometry: sign and orthogonality
  • Vector orthogonality
  • Relative vector angles
  • Code challenge: dot product sign and scalar multiplication
  • Code challenge: is the dot product commutative?
  • Vector Hadamard multiplication
  • Outer product
  • Vector cross product
  • Vectors with complex numbers
  • Hermitian transpose (a.k.a. conjugate transpose)
  • Interpreting and creating unit vectors
  • Code challenge: dot products with unit vectors
  • Dimensions and fields in linear algebra
  • Subspaces
  • Subspaces vs. subsets
  • Span
  • In the span?
  • Linear independence
  • Basis
Introduction to matrices
  • Exercises + code
  • Matrix terminology and dimensionality
  • Matrix sizes and dimensionality
  • A zoo of matrices
  • Can the matrices be concatenated?
  • Matrix addition and subtraction
  • Matrix-scalar multiplication
  • Code challenge: is matrix-scalar multiplication a linear operation?
  • Transpose
  • Complex matrices
  • Addition, equality, and transpose
  • Diagonal and trace
  • Code challenge: linearity of trace
  • Broadcasting matrix arithmetic
Matrix multiplications
  • Exercises + code
  • Introduction to standard matrix multiplication
  • Four ways to think about matrix multiplication
  • Code challenge: matrix multiplication by layering
  • Matrix multiplication with a diagonal matrix
  • Order-of-operations on matrices
  • Matrix-vector multiplication
  • Find the missing value!
  • 2D transformation matrices
  • Code challenge: Pure and impure rotation matrices
  • Code challenge: Geometric transformations via matrix multiplications
  • Additive and multiplicative matrix identities
  • Additive and multiplicative symmetric matrices
  • Hadamard (element-wise) multiplication
  • Matrix operation equality
  • Code challenge: symmetry of combined symmetric matrices
  • Multiplication of two symmetric matrices
  • Code challenge: standard and Hadamard multiplication for diagonal matrices
  • Code challenge: Fourier transform via matrix multiplication!
  • Frobenius dot product
  • Matrix norms
  • Code challenge: conditions for self-adjoint
  • What about matrix division?
Matrix rank
  • Exercises + code
  • Rank: concepts, terms, and applications
  • Maximum possible rank.
  • Computing rank: theory and practice
  • Rank of added and multiplied matrices
  • What's the maximum possible rank?
  • Code challenge: reduced-rank matrix via multiplication
  • Code challenge: scalar multiplication and rank
  • Rank of A^TA and AA^T
  • Code challenge: rank of multiplied and summed matrices
  • Making a matrix full-rank by "shifting"
  • Code challenge: is this vector in the span of this set?
  • Course tangent: self-accountability in online learning
Matrix spaces
  • Exercises + code
  • Column space of a matrix
  • Column space, visualized in code
  • Row space of a matrix
  • Null space and left null space of a matrix
  • Column/left-null and row/null spaces are orthogonal
  • Dimensions of column/row/null spaces
  • Example of the four subspaces
  • More on Ax=b and Ax=0