Column space

In linear algebra, the column space (or range) 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 ${\displaystyle \mathbb {F} }$ be a field. The column space of an m × n matrix with components from ${\displaystyle \mathbb {F} }$ is a linear subspace of the m-space ${\displaystyle \mathbb {F} ^{m}}$. The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring ${\displaystyle \mathbb {K} }$ is also possible.

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