# Linear algebra

**Linear algebra** is the branch of mathematics that deals with vectors, vector spaces, linear maps, and systems of linear equations. Topics studied in Linear algebra include Linear equations, Matrices, Matrix decompositions, Computations, Vectors, Vector spaces, Multilinear algebra, Affine space, Projective space.^{[1]}
Linear algebra has numerous applications in engineering, chemistry and physics. Matrices can be used for data fitting tasks such as "curving" student grades on exams, and for finding correlations between two sets of data. Linear algebra is particularly useful for organizing and simplifying data.

The three most vexing computations in linear algebra are these:

- linear equations
- least squares (data fitting)
- finding the eigenvalues of an
*n x n*matrix for n > 3.

## Contents

## Basic Concepts

- transformations, inverses and matrix products

- Subspaces

- 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)

- eigenbasis
- orthogonal eigenbasis
- orthonormal eigenbasis

- diagonalization
- trace
- characteristic polynomial
- complex eigenvalues

- 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

- Linear dynamical systems

- Euler's Formula

### Notation

- matrices are commonly represented by
*A*and*B* - diagonal matrices are represented by
*D* - an upper triangular matrix is represented by
*R*, as in QR factorization - when
*A*is similar to*B*, then an invertible matrix*S*is used to represent that*AS*=*SB*

## 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:

- find the characteristic polynomial
- solve for the eigenvalues
- solve for the eigenvectors

- finding the orthogonal projection of a vector in a vector space
- 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
- QR factorization
- find the least squares solution for a set of data
- use known characteristics of symmetric or diagonalizable matrices to find solutions

## Application: Analyzing Liberal style on Wikipedia

An *n x m* matrix can be developed using observed incidents of liberal style in Wikipedia entries, and that data can then be simplified to draw conclusions about how liberal style can mislead viewers. The *n* rows can represent different elements of liberal style, while the *m* columns can represent different types of entries on Wikipedia.

The prevalence of certain types of liberal style may reflect their perceived effectiveness, and a linear algebra-based approach at modeling it may be useful in debunking the flawed reasoning and claims.