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.

Basic Concepts

• Linear equations

• coefficient matrices and Gauss-Jordan elimination
• geometric representations, especially vectors

• Linear transformations

• transformations, inverses and matrix products

• Subspaces

• image and kernel
• basis and span
• dimension

• Determinants

• geometrical interpretation
• 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
• complex eigenvalues

• Orthogonality

• 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)

• 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
• 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