.01

.02

.03

• Operations on one matrix

  • Linear systems in two unknowns

  • Linear systems in three unknowns

  • Matrix dimensions and entries

  • Representing systems with matrices

  • Simple row operations

  • Pivot entries and row-echelon forms

  • Gauss-Jordan elimination

  • Number of solutions to the linear system

• Operations on two matrices

  • Matrix addition and subtraction

  • Scalar multiplication

  • Zero matrices

  • Matrix multiplication

  • Identity matrices

  • The elimination matrix

• Matrices as vectors

  • Vectors

  • Vector operations

  • Unit vectors and basis vectors

  • Linear combinations and span

  • Linear independence in two dimensions

  • Linear independence in three dimensions

  • Linear subspaces

  • Spans as subspaces

  • Basis

• Dot products and cross products

  • Dot products

  • Cauchy-Schwarz inequality

  • Vector triangle inequality

  • Angle between vectors

  • Equation of a plane, and normal vectors

  • Cross products

  • Dot and cross products as opposite ideas

• Matrix-vector products

  • Multiplying matrices by vectors

  • The null space and Ax=O

  • Null space of a matrix

  • The column space and Ax=b

  • Solving Ax=b

  • Dimensionality, nullity, and rank

• Transformations

  • Functions and transformations

  • Transformation matrices and the image of the subset

  • Preimage, image, and the kernel

  • Linear transformations as matrix-vector products

  • Linear transformations as rotations

  • Adding and scaling linear transformations

  • Projections as linear transformations

  • Compositions of linear transformations

• Inverses

  • Inverse of a transformation

  • Invertibility from the matrix-vector product

  • Inverse transformations are linear

  • Matrix inverses, and invertible and singular matrices

  • Solving systems with inverse matrices

• Determinants

  • Determinants

  • Cramer's rule for solving systems

  • Modifying determinants

  • Upper and lower triangular matrices

  • Using determinants to find area

• Transposes

  • Transposes and their determinants

  • Transposes of products, sums, and inverses

  • Null and column spaces of the transpose

  • The product of a matrix and its transpose

• Orthogonality and change of basis

  • Orthogonal complements

  • Orthogonal complements of the fundamental subspaces

  • Projection onto the subspace

  • Least squares solution

  • Coordinates in a new basis

  • Transformation matrix for a basis

• Orthonormal bases and Gram-Schmidt

  • Orthonormal bases

  • Projection onto an orthonormal basis

  • Gram-Schmidt process for change of basis

• Eigenvalues and Eigenvectors
  

  • Eigenvalues, Eigenvectors, Eigenspaces

  • Eigen in three dimensions

.04

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