Read the unit, then let the drill mark you.

Systems of Linear Equations

Spotting a linear equation, writing a solution set in parameters, telling consistent from inconsistent, the augmented matrix and the three row operations, row-echelon and reduced row-echelon form, and solving by Gaussian and Gauss-Jordan elimination.

Operations with Matrices

When two matrices are equal, adding and scaling them, when a product is defined and what size it is, row-times-column multiplication, a system as Ax = b and a product as a combination of columns, the algebra rules that do and don't hold, and the transpose.

The Inverse of a Matrix

What makes B the inverse of A, the ad − bc formula for a 2 × 2, finding A⁻¹ by row-reducing [A | I], the rules for (AB)⁻¹, (Aᵀ)⁻¹ and (cA)⁻¹, and solving a system as x = A⁻¹b — the idea behind the course's Hill-cipher activity.

Determinants

ad − bc for a 2 × 2, the minor Mᵢⱼ and the cofactor Cᵢⱼ with its checkerboard sign, expanding along the row or column with the most zeros for a 3 × 3 or 4 × 4, and the triangular shortcut: multiply the main diagonal.

Vector Spaces

Vectors in Rⁿ and their sum, scalar multiple and negative, writing one vector as a linear combination of others, the ten axioms that make a set a vector space and the one that breaks, and testing whether a subset is a subspace.

Unit 1 · Module 1 · Exam 1 · §1.1–1.2

Systems of Linear Equations

Unit 2 · Module 2 · Exam 1 · §2.1–2.2

Operations with Matrices

Unit 3 · Module 2 · Exam 1 · §2.3

The Inverse of a Matrix

Unit 4 · Module 3 · §3.1

Determinants

Unit 5 · Module 4 · Exam 2 · §4.1–4.3

Vector Spaces