This is a glossary of linear algebra.

See also: glossary of module theory.

- Affine transformation
- A composition of functions consisting of a linear transformation between vector spaces followed by a translation.
^{[1]}Equivalently, a function between vector spaces that preserves affine combinations. - Affine combination
- A linear combination in which the sum of the coefficients is 1.

- Basis
- In a vector space, a linearly independent set of vectors spanning the whole vector space.
^{[2]} - Basis vector
- An element of a given basis of a vector space.
^{[2]}

- Column vector
- A matrix with only one column.
^{[3]} - Coordinate vector
- The tuple of the coordinates of a vector on a basis.
- Covector
- An element of the dual space of a vector space, (that is a linear form), identified to an element of the vector space through an inner product.

- Determinant
- The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of for the unit matrix.
- Diagonal matrix
- A matrix in which only the entries on the main diagonal are non-zero.
^{[4]} - Dimension
- The number of elements of any basis of a vector space.
^{[2]} - Dual space
- The vector space of all linear forms on a given vector space.
^{[5]}

- Elementary matrix
- Square matrix that differs from the identity matrix by at most one entry

- Identity matrix
- A diagonal matrix all of the diagonal elements of which are equal to .
^{[4]} - Inverse matrix
- Of a matrix , another matrix such that multiplied by and multiplied by both equal the identity matrix.
^{[4]} - Isotropic vector
- In a vector space with a quadratic form, a non-zero vector for which the form is zero.
- Isotropic quadratic form
- A vector space with a quadratic form which has a null vector.

- Linear algebra
- The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.
- Linear combination
- A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element).
^{[6]} - Linear dependence
- A linear dependence of a tuple of vectors is a nonzero tuple of scalar coefficients for which the linear combination equals .
- Linear equation
- A polynomial equation of degree one (such as ).
^{[7]} - Linear form
- A linear map from a vector space to its field of scalars
^{[8]} - Linear independence
- Property of being not linearly dependent.
^{[9]} - Linear map
- A function between vector spaces which respects addition and scalar multiplication.
- Linear transformation
- A linear map whose domain and codomain are equal; it is generally supposed to be invertible.

- Matrix
- Rectangular arrangement of numbers or other mathematical objects.
^{[4]}

- Null vector
- 1. Another term for an isotropic vector.
- 2. Another term for a zero vector.

- Row vector
- A matrix with only one row.
^{[4]}

- Singular-value decomposition
- a factorization of an complex matrix
**M**as , where**U**is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, and**V**is an complex unitary matrix.^{[10]} - Spectrum
- Set of the eigenvalues of a matrix.
^{[11]} - Square matrix
- A matrix having the same number of rows as columns.
^{[4]}

- Unit vector
- a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.
^{[12]}

- Vector
- 1. A directed quantity, one with both magnitude and direction.
- 2. An element of a vector space.
^{[13]} - Vector space
- A set, whose elements can be added together, and multiplied by elements of a field (this is scalar multiplication); the set must be an abelian group under addition, and the scalar multiplication must be a linear map.
^{[14]}

- Zero vector
- The additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.
^{[15]}