Linear algebra

Linear algebra is the mathematical subject that studies vectors, vector spaces, linear maps, and systems of linear equations. Branches of Linear algebra typically include Linear equations, Matrices, Matrix decompositions, Computations, Vectors, Vector spaces, Multilinear algebra, Affine space, Projective space. Linear algebra has numerous applications in engineering, chemistry and physics.

The three most vexing computations in linear algebra are these:

1. linear equations
2. least squares
3. finding the eigenvalues of an n x n matrix for n > 3.

Basic Concepts

• Linear equations

• coefficient matrices and Gauss-Jordan elimination
• rank (row rank = column rank)
• geometric representations, especially vectors and systems with more variables than equations

• Linear transformations

• transformations, inverses and matrix products

• Subspaces

• image and kernel
• basis and span
• dimension

• Determinants

• Sarrus's rule
• geometrical interpretation

• classical adjoint, expansion factor, application to parallelepipeds)

• determinant of similar matrix, inverse matrix, product of matrices
• Cramer's rule (with and without product rule)
• minor of a matrix
• Laplace expansion (cofactors)

• Basis

• eigenbasis
• orthogonal eigenbasis
• orthonormal eigenbasis

• Eigenvalues and eigenvectors

• diagonalization
• trace
• characteristic polynomial
• complex eigenvalues

• Orthogonality

• orthogonal (perpendicular) vectors
• orthonormal vectors
• orthogonal projections
• orthogonal matrix
• projections
• Gram-Schmidt Process and QR Factorization
• orthogonal matrices, orthogonal transformations
• data fitting, especially least squares

• Symmetric matrix

• spectral theorem

• Quadratic forms

• Linear dynamical systems

• Euler's Formula

More advanced topics include

• Vector spaces (linear spaces)

• the conditions of a vector space
• isomorphisms
• Nth dimensional spaces and subspaces

• Inner spaces

• inner product spaces

• Determinants

• cofactor
• adjugate (useful in finding the inverse of a matrix)

• Cayley-Hamilton theorem

• Stability

• Hermitian Matrices

• Singular values and Singular Value Decomposition

• Linear differential equations

Common problems

Common problems in linear algebra include:

• simplifying or reducing matrices
• Gauss-Jordan elimination
• matrix multiplication
• finding inverses and transposes of matrices
• Gram-Schmidt procedure
• finding eigenvalues and eigenvectors for matrices:

1. find the characteristic polynomial
2. solve for the eigenvalues
3. solve for the eigenvectors

• diagonalize a matrix
• find the geometric equivalent of a matrix
• finding the determinant of a 2x2 matrix (easy) and a 3x3 matrix (hard)
• finding the inverse of a matrix
• decomposition or factorization of a matrix: representing a given matrix as a product of simpler matrices