Linear Algebra Tags

Systems of linear equations

• Systems with 2 variables
• Systems with 3 variables
• Systems with 4 or more variables
• Matrix-vector forms
• Vector equations
• Augmented matrices

Matrices

• Matrix basics
• Matrix algebra
• Row operations
• Echelon form
• Rank
• Transpose and trace
• Inverses
• Elementary matrices
• Complex entries
• Markov chains

Matrix factorizations

• Diagonalization
• LU factorization
• QR factorization
• Singular value decomposition
• Jordan form

Euclidean spaces

• Vectors
• Linear combinations
• Span
• Linear independence
• Subspaces
• Basis and dimension
• Row, column, and null spaces
• Coordinate vectors and change of basis

Vector geometry

• Vectors and vector arithmetic
• Dot product, length, and unit vectors
• Cross product
• Planes
• Lines with planes
• Coordinate systems

Abstract vector spaces

• Definition and properties
• Linear combinations
• Span
• Linear independence
• Subspaces
• Basis and dimension
• Coordinate vectors and change of basis
• Examples

Eigenvalues and eigenvectors

• Computing eigenvalues and eigenvectors
• Properties
• Complex eigenvalues and eigenvectors
• Quadratic forms

Inner products

• Computing with dot products
• Computing with inner products
• Orthogonal and orthonormal sets
• Projection and distance
• Gram-Schmidt process
• Orthogonal matrices

Linear transformations

• Evaluating linear transformations
• Associated matrices
• One-to-one and onto
• Kernel and image
• Inverses

Determinants

• Computing determinants
• Properties
• Applications