- 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

- 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

- 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

- 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?

- 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

- 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