Row Space ○｜Definition｜1st｜20260122122521-00-⌔

Row and column spaces - Wikipedia

Row and column spaces

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In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.

Let $F$ be a field. The column space of an m × n matrix with components from $F$ is a linear subspace of the m -space $F^m$. The dimension of the column space is called the rank of the matrix and is at most min(m, n).¹ A definition for matrices over a ring $R$ is also possible.

The row space is defined similarly.

The row space and the column space of a matrix A are sometimes denoted as C (A${}^T$) and C (A) respectively.²

This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$ respectively.³

Printed 2026-06-28.

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Footnotes

1. Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang 2005. ↩

2. Strang, Gilbert (2016). Introduction to linear algebra (Fifth ed.). Wellesley, MA: Wellesley-Cambridge Press. pp. 128, 168. ISBN 978-0-9802327-7-6. OCLC 956503593. ↩

3. Anton (1987, p. 179) ↩