Column or Row Space Analyzer

Math Linear Algebra • Vector Spaces and Linear Independence

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Given a matrix \(A\in\mathbb{R}^{m\times n}\), this tool finds a basis for: Col(A) (column space / image) using pivot columns of the original \(A\), and Row(A) (row space) using the non-zero rows of RREF(A). Both spaces have dimension equal to the rank.

Rows \(m\)

Columns \(n\)

Tolerance

Display decimals

Steps

Show row-ops log Orthonormalize basis (Gram–Schmidt) Plot Col(A) when \(m=2\) Plot Row(A) when \(n=2\)

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Matrix \(A\)

A is 2×3

Column basis = pivot columns of \(A\). Row basis = non-zero rows of RREF(\(A\)).

Results

Rank

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\(\dim(\mathrm{Col}(A))\)

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\(\dim(\mathrm{Row}(A))\)

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Pivot columns

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Basis for \(\mathrm{Col}(A)\)

These are the pivot columns taken from the original matrix \(A\).

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Basis for \(\mathrm{Row}(A)\)

These are the non-zero rows of \(\mathrm{rref}(A)\) (a standard row-space basis).

—

Ready

RREF of \(A\)

Pivot columns of \(A\) are determined from pivot positions in RREF.

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Col(A) plot (when \(m=2\))

Column space lives in \(\mathbb{R}^m\). In 2D, it is \(\{0\}\), a line, or all of \(\mathbb{R}^2\).

Drag to pan • wheel to zoom • double-click to reset view

Row(A) plot (when \(n=2\))

Row space lives in \(\mathbb{R}^n\). In 2D, it is \(\{0\}\), a line, or all of \(\mathbb{R}^2\).

Drag to pan • wheel to zoom • double-click to reset view

Enter \(A\) and click “Calculate”.

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Theory

Column space and row space

For a matrix \(A\in\mathbb{R}^{m\times n}\):

The column space is \[ \mathrm{Col}(A)=\mathrm{span}\{\text{columns of }A\}\subseteq \mathbb{R}^{m}. \] It is also called the image of the linear map \(x\mapsto Ax\).
The row space is \[ \mathrm{Row}(A)=\mathrm{span}\{\text{rows of }A\}\subseteq \mathbb{R}^{n}. \]

Both are subspaces, so they have well-defined dimensions and bases.

Rank and dimensions

The rank of \(A\), written \(\mathrm{rank}(A)\), equals:

the number of pivot positions in \(\mathrm{rref}(A)\)
\(\dim(\mathrm{Col}(A))\)
\(\dim(\mathrm{Row}(A))\)

Therefore, \[ \dim(\mathrm{Col}(A))=\dim(\mathrm{Row}(A))=\mathrm{rank}(A). \]

How to get a basis for the column space

Compute \(R=\mathrm{rref}(A)\) and identify the pivot columns (the columns containing leading 1s). If the pivot columns are \(j_1,\dots,j_r\), then a basis for the column space is: \[ \{A_{(:,j_1)},\ A_{(:,j_2)},\ \dots,\ A_{(:,j_r)}\}, \] meaning the corresponding columns taken from the original matrix \(A\), not from \(R\).

Why original columns? Row operations change columns but do not change which original columns are needed to generate \(\mathrm{Col}(A)\).

How to get a basis for the row space

Row operations do not change the row space, so \(A\) and \(R=\mathrm{rref}(A)\) have the same row space. The non-zero rows of \(R\) are automatically linearly independent and they span \(\mathrm{Row}(A)\). Therefore, a standard row-space basis is simply the list of non-zero rows of \(R\).

University note: orthonormal bases

Sometimes you want an orthonormal basis (vectors of length 1 that are mutually orthogonal). Given any independent basis, you can produce an orthonormal basis using the Gram–Schmidt process. This is useful for projections, least squares, and numerical stability.

The calculator includes an optional Gram–Schmidt toggle to preview orthonormal bases for \(\mathrm{Col}(A)\) and \(\mathrm{Row}(A)\).

Geometric intuition in 2D

When the column space lives in \(\mathbb{R}^2\) (i.e., \(m=2\)), it can only be:

\(\{0\}\) (rank 0)
a line through the origin (rank 1)
all of \(\mathbb{R}^2\) (rank 2)

Similarly, when \(n=2\), the row space is a subspace of \(\mathbb{R}^2\) with the same possibilities.

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