Span and Basis Finder

Math Linear Algebra • Vector Spaces and Linear Independence

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

Given vectors \(v_1,\dots,v_k\in\mathbb{R}^d\), this tool finds the dimension of the span and reduces the set to a basis subset using elimination (RREF pivot columns). If a vector is redundant, it also shows a relation and the redundancy form: \(v_j=\sum \alpha_t v_{i_t}\) and \(v_j-\sum \alpha_t v_{i_t}=0\).

Dimension \(d\)

Number of vectors \(k\)

Input orientation

Tolerance

Display decimals

Steps

Show removed-vector relations Simplify relation coefficients to integers (when possible) Show 2D span plot when \(d=2\) Show nonnegative cone (2D)

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. The basis is chosen as the earliest pivot vectors (left-to-right pivots).

Vector matrix input

A is 2×3 (columns are vectors)

Pivot columns correspond to basis vectors.

Results

Span dimension \(\dim(\mathrm{span}\{v_i\})\)

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Spans \(\mathbb{R}^d\)?

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Basis vectors (subset indices)

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Minimal basis size

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Basis (as original vectors)

These vectors span the same subspace as the full set.

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Ready

RREF and pivot columns

Pivot columns indicate which original vectors form a basis for the span (column space).

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Enter vectors and click “Calculate”.

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Theory

Span and spanning sets

The span of vectors \(v_1,\dots,v_k\) is the set of all linear combinations: \[ \mathrm{span}\{v_1,\dots,v_k\}=\left\{\sum_{i=1}^k c_i v_i \;:\; c_i\in\mathbb{R}\right\}. \] A set of vectors is a spanning set for a subspace \(W\) if every vector in \(W\) can be written as a linear combination of them.

Many spanning sets contain redundancy: some vectors can be written using others, and removing them does not change the span.

Basis and “minimal generators”

A basis for a subspace \(W\) is a set of vectors that:

Spans \(W\), and
Is linearly independent (no vector is redundant).

Because of independence, a basis is a minimal generating set: removing any basis vector changes the span. The number of vectors in any basis of \(W\) is the dimension of \(W\), written \(\dim(W)\).

Matrix method: pivot columns give a basis

Put the vectors into a matrix as columns: \[ A=[v_1\ v_2\ \cdots\ v_k]\in\mathbb{R}^{d\times k}. \] The column space of \(A\) is exactly \(\mathrm{span}\{v_1,\dots,v_k\}\).

Compute the reduced row echelon form (RREF) of \(A\). The columns of \(A\) that correspond to pivot columns form a basis for the column space. The number of pivots is the rank: \[ \dim(\mathrm{span}\{v_i\})=\text{rank}(A). \]

If \(\text{rank}(A)=d\), then the vectors span all of \(\mathbb{R}^d\). If \(\text{rank}(A)<d\), they span a proper subspace.

How the tool builds “removed vector” relations

Once a basis subset \(\{v_{i_1},\dots,v_{i_r}\}\) is chosen, every vector in the span can be written uniquely as \[ v = \alpha_1 v_{i_1} + \cdots + \alpha_r v_{i_r}. \] The calculator finds the coefficients \(\alpha\) by solving the linear system \(A_{\text{basis}}\alpha=v\), where \(A_{\text{basis}}\) contains only the basis columns.

When a vector \(v_j\) is not a pivot vector, the tool reports one such relation. For example, if \(v_2=2v_1\), then \(v_2\) is redundant and can be removed without changing the span.

Geometric intuition (2D “span cone”)

In \(\mathbb{R}^2\), a single nonzero vector spans a line through the origin. Two non-collinear vectors span all of \(\mathbb{R}^2\). The calculator can draw the basis vectors and (optionally) a small region representing nonnegative combinations of the basis, which is often called a cone in geometry and optimization.

University note: function spaces and infinite dimensions

Span and basis also apply in function spaces (e.g., polynomials, Fourier series) where the “vectors” are functions. In infinite-dimensional spaces, bases can be more subtle (Hamel bases vs orthonormal bases), and computational methods typically rely on inner products, orthogonality, and projections rather than finite elimination.

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