• The Space R²
• The Space R³
• The Space Rⁿ

• Matrices
• Operations with Matrices

• Introduction to Linear Systems
• Solutions to Linear Systems
• Gaussian Elimination
• Using Elementary Row Operations to Determine A⁻¹

• Subspaces of Rⁿ
• The Nullspace of a Matrix
• Linear Combinations and Span
• Linear Independence
• The Rank of a Matrix
• A Basis for a Vector Space
• Projection onto a Subspace
• Row Space and Column Space of a Matrix
• The Rank Plus Nullity Theorem
• More Vector Spaces; Isomorphism

• Definitions of the Determinant
• Laplace Expansions for the Determinant
• Cramer's Rule
• The Classical Adjoint of a Square Matrix

• Eigenvalue and Eigenvector Defined
• Determining the Eigenvalues of a Matrix
• Determining the Eigenvectors of a Matrix
• Eigenspaces
• Diagonalization