Linear Independence Checker

Math Linear Algebra • Vector Spaces and Linear Independence

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

Test whether vectors \(v_1,\dots,v_k\in\mathbb{R}^d\) are linearly independent. The checker uses RREF (pivot count / rank) and, if dependent, returns one non-trivial coefficient vector \(c\neq 0\) such that \(\sum_{i=1}^k c_i v_i = 0\).

Dimension \(d\)

Number of vectors \(k\)

Input orientation

Tolerance

Display decimals

Steps

Show dependence coefficients (if dependent) Show 2D plot when \(d=2\) Simplify coefficients to integers (when possible)

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. Rule of thumb: if \(k > d\), the vectors are automatically dependent.

Vector matrix input

A is 2×3 (columns are vectors)

Each column is a vector \(v_i\) when “Vectors as columns” is selected.

Results

Independence

—

Rank (pivot count)

—

Nullity \(=k-\text{rank}\)

—

Basis candidate?

—

Ready

RREF and pivots

Vectors are independent iff the number of pivots equals \(k\) (and necessarily \(k\le d\)).

—

Enter vectors and click “Calculate”.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory

Linear independence (definition)

Vectors \(v_1,\dots,v_k\) in a vector space are linearly independent if the only solution to \[ c_1v_1+\cdots+c_kv_k=0 \] is the trivial solution \(c_1=\cdots=c_k=0\). If there exists a non-trivial coefficient vector \(c\neq 0\) that makes the combination zero, the set is linearly dependent.

Matrix test using RREF and rank

Put the vectors into a matrix \[ A = [v_1\ v_2\ \cdots\ v_k] \] as columns. Then the equation \(c_1v_1+\cdots+c_kv_k=0\) becomes a single matrix equation \[ Ac = 0, \] where \(c=(c_1,\dots,c_k)^T\).

The set is independent iff the only solution to \(Ac=0\) is \(c=0\), which happens exactly when \[ \text{nullity}(A)=0 \quad\Longleftrightarrow\quad \text{rank}(A)=k. \] In RREF language: independent iff every column is a pivot column.

Rank–nullity and the “k > d” shortcut

If \(A\) is a \(d\times k\) matrix, then \[ \text{rank}(A)\le \min(d,k). \] Therefore, if you have more vectors than the dimension (\(k>d\)), then \(\text{rank}(A)\le d < k\), so the vectors are automatically dependent.

Rank–nullity gives the precise count of degrees of freedom in the dependence: \[ \text{rank}(A)+\text{nullity}(A)=k. \] When nullity is positive, there are infinitely many dependence relations; the calculator returns one example \(c\neq 0\).

Geometric intuition (2D and 3D)

In \(\mathbb{R}^2\), two nonzero vectors are independent exactly when they are not collinear (not scalar multiples). Three vectors in \(\mathbb{R}^2\) must be dependent.

In \(\mathbb{R}^3\), three vectors are independent when they do not lie in the same plane through the origin. More than three vectors in \(\mathbb{R}^3\) must be dependent.

Basis candidate check

A set of vectors is a basis for \(\mathbb{R}^d\) if it is independent and spans \(\mathbb{R}^d\). For a finite set of \(k\) vectors in \(\mathbb{R}^d\), a common quick test is:

If \(k=d\) and the vectors are independent, they form a basis of \(\mathbb{R}^d\).
If \(k<d\), they cannot span \(\mathbb{R}^d\) even if they are independent.
If \(k>d\), they cannot be independent.

University note: infinite-dimensional spaces

Linear independence extends beyond \(\mathbb{R}^d\) to function spaces, polynomial spaces, and more. In infinite-dimensional settings, independence is still defined by “no non-trivial finite linear combination equals zero,” but the computational tools change (e.g., orthogonality, inner products, and functional analysis methods).

Rate this calculator

Related calculators