Linear Combination Solver

Math Linear Algebra • Vector Spaces and Linear Independence

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

Find coefficients \(c_1,\dots,c_k\) such that \[ v = \sum_{i=1}^{k} c_i u_i. \] Put \(u_i\) as columns of \(U=[u_1\ \cdots\ u_k]\). Then solve \(Uc=v\). This tool checks consistency via ranks, returns one solution (or a minimum-norm solution when multiple exist), and can optionally show a least-squares approximation when there is no exact solution.

Dimension \(d\)

Number of vectors \(k\)

Vector orientation

Solution preference

Tolerance

Display decimals

Steps

Show row-ops log (RREF) Show parametric form (if infinite solutions) Simplify coefficients to integers (when possible) If no exact solution: show least-squares approximation Show 2D weighted-sum animation when \(d=2\)

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

Vectors \(u_1,\dots,u_k\)

U is 2×2

We solve \(Uc=v\). Columns are \(u_i\) (internally).

Target vector \(v\)

Enter the vector you want to express as a linear combination.

Results

Exact solution?

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rank(U)

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rank([U|v])

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Nullity (k − rank)

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Coefficients \(c\)

If solutions are infinite, this returns one solution (or the minimum-norm solution if selected).

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Check

Shows \(Uc\), residual \(Uc-v\), and \(\|Uc-v\|\).

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Ready

RREF panels

Exact solvability uses the rank test: \(Uc=v\) is solvable iff \(\mathrm{rank}(U)=\mathrm{rank}([U|v])\).

RREF(U)

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RREF([U | v])

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Enter vectors and click “Calculate”.

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Theory

Linear combination and span

A vector \(v\in\mathbb{R}^d\) is a linear combination of vectors \(u_1,\dots,u_k\) if there exist scalars \(c_1,\dots,c_k\) such that \[ v = c_1u_1 + c_2u_2 + \cdots + c_ku_k. \] The set of all such combinations is the span: \[ \mathrm{span}\{u_1,\dots,u_k\}=\{c_1u_1+\cdots+c_ku_k:\ c_i\in\mathbb{R}\}. \] This span is a subspace of \(\mathbb{R}^d\).

Matrix form \(Uc=v\)

Put the vectors as columns of a matrix \[ U=[u_1\ u_2\ \cdots\ u_k]\in\mathbb{R}^{d\times k}, \] and collect coefficients into \(c=[c_1,\dots,c_k]^T\). Then \[ v=\sum_{i=1}^k c_i u_i \quad \Longleftrightarrow \quad Uc=v. \] So finding coefficients is a linear system-solving problem.

When does a solution exist?

The equation \(Uc=v\) has a solution exactly when \(v\) lies in the column space of \(U\): \[ v\in \mathrm{Col}(U)=\mathrm{span}\{u_1,\dots,u_k\}. \] A practical test uses ranks: \[ Uc=v \text{ is solvable } \iff \mathrm{rank}(U)=\mathrm{rank}([U|v]). \] In RREF, inconsistency appears as a row of the form \([0\ \cdots\ 0\ |\ \beta]\) with \(\beta\neq 0\).

Uniqueness vs infinite solutions

If a solution exists, it can be:

Unique when \(U\) has full column rank (\(\mathrm{rank}(U)=k\)).
Infinitely many when \(\mathrm{rank}(U)<k\). Then there are free variables.

The number of degrees of freedom is the nullity: \[ \mathrm{nullity}(U)=k-\mathrm{rank}(U)=\dim(\mathrm{Null}(U)). \] All solutions have the form \[ c=c_0 + N t, \] where \(c_0\) is a particular solution, columns of \(N\) form a basis of \(\mathrm{Null}(U)\), and \(t\) is a parameter vector.

Minimum-norm solution (university)

When solutions are not unique, a common canonical choice is the solution with smallest coefficient length: \[ \min \|c\| \quad \text{subject to } Uc=v. \] This is the minimum-norm solution. The calculator can compute it by adjusting a particular solution within the null space.

If no exact solution: least squares approximation

If \(v\notin\mathrm{span}\{u_i\}\), the system is inconsistent. A standard alternative is the least-squares problem: \[ \min_c \|Uc-v\|. \] The resulting \(Uc\) is the closest vector to \(v\) that lies in the column space of \(U\) (a projection onto the span). The calculator can optionally show this approximation and the residual distance.

Special case: \(v=0\)

The equation \(Uc=0\) always has the trivial solution \(c=0\). If the vectors \(u_i\) are linearly dependent, there are also non-trivial solutions \(c\neq 0\), meaning the null space of \(U\) is non-zero.

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