Column space

related topics
{math, number, function}

In linear algebra, the column space of a matrix (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of m-dimensional Euclidean space. The dimension of the column space is called the rank of the matrix.[1]

The column space of a matrix is the image or range of the corresponding matrix transformation.

Contents

Definition

Let A be an m × n matrix, with column vectors v1v2, ..., vn. A linear combination of these vectors is any vector of the form

where c1c2, ..., cn are scalars. The set of all possible linear combinations of v1,...,vn is called the column space of A. That is, the column space of A is the span of the vectors v1,...,vn.

Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:

Therefore, the column space of A consists of all possible products Ax, for x ∈ Rn. This is the same as the image (or range) of the corresponding matrix transformation.

Basis

The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space.

For example, consider the matrix

The columns of this matrix span the column space, but they may not be linearly independent, in which case some subset of them will form a basis. To find this basis, we reduce A to reduced row echelon form:

Full article ▸

related documents
Monotone convergence theorem
Operator overloading
Quasigroup
Special linear group
Borel algebra
Automata theory
Graftal
Golden ratio base
Generating trigonometric tables
Least common multiple
Associative algebra
Stirling's approximation
Sylow theorems
1 (number)
Jacobi symbol
Theory of computation
Measure (mathematics)
Product topology
Spectrum of a ring
Free variables and bound variables
Diophantine equation
Mean value theorem
De Morgan's laws
Cumulative distribution function
Perfect number
Chain complex
Elliptic integral
Discriminant
Cauchy-Riemann equations
NaN