"Linear algebra is the branch of mathematics concerning linear equations..."
The study of matrices, vectors, and linear equations, including linear transformations and matrix algebra.
Vectors: A vector is a mathematical object that has both magnitude and direction. In linear algebra, vectors are usually represented as column matrices.
Matrices: A matrix is a rectangular array of numbers. Linear algebra heavily relies on matrices to represent linear transformations.
Matrix Operations: Basic matrix operations include addition, multiplication, and scalar multiplication. More advanced operations include inversion, determinant, and eigenvalues.
Systems of Linear Equations: A system of linear equations is a collection of linear equations that are meant to be solved simultaneously. The solution to a system of linear equations can be represented as a column matrix.
Linear Transformations: A linear transformation is a function that maps vectors from one space to another. Linear transformations can be represented by matrices, such as rotation, reflection, and scaling.
Vector Spaces: A vector space is a collection of vectors that satisfy certain properties. A vector space is closed under vector addition and scalar multiplication.
Basis and Dimension: A basis is a set of linearly independent vectors that span a vector space. The dimension of a vector space is the number of vectors in its basis.
Orthogonality: Orthogonality refers to the relationship between two vectors that are perpendicular to each other. Orthogonal vectors have a dot product of zero.
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are used to describe the behavior of linear transformations. Eigenvalues are scalars, and eigenvectors are vectors that point in the direction that does not change under the transformation.
Inner Product Spaces: An inner product space is a vector space with an additional inner product operation. This operation allows for the definition of distance, angles, and orthogonality in the vector space.
Least Squares Approximation: Least squares approximation is a method for solving an over-determined system of linear equations. It finds the vector that minimizes the sum of the squares of the differences between the equations and the corresponding values.
Singular Value Decomposition: Singular value decomposition is a factorization of a matrix that expresses it as the product of three matrices. It is useful for many applications, including image compression, data mining, and regression analysis.
"For instance, a₁x₁ + ... + aₙxₙ = b can be considered a linear equation."
"Linear maps such as (x₁, ..., xₙ) ↦ a₁x₁ + ... + aₙxₙ..."
"...their representations in vector spaces and through matrices."
"Linear algebra is central to almost all areas of mathematics."
"Linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes, and rotations."
"Functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions."
"Linear algebra is also used in most sciences and fields of engineering..."
"...because it allows modeling many natural phenomena..."
"...and computing efficiently with such models."
"For nonlinear systems, which cannot be modeled with linear algebra..."
"...it is often used for dealing with first-order approximations..."
"...using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point."
"For instance, a₁x₁ + ... + aₙxₙ = b can be considered a linear equation."
"Linear algebra is also used in most sciences and fields of engineering..."
"Linear algebra is also used in most sciences and fields of engineering..."
"Linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes, and rotations."
"Functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions."
"For nonlinear systems, which cannot be modeled with linear algebra..."
"...it is often used for dealing with first-order approximations..."