Linear Combination Calculator

A linear combination of vectors v₁, v₂, ..., vₙ is an expression of the form c₁v₁ + c₂v₂ + ... + cₙvₙ where c₁, c₂, ..., cₙ are scalar coefficients. Any vector in the span of the basis vectors can be written as such a combination.

Set up a system of linear equations by equating each component of the target vector to the corresponding linear combination of basis vector components. Solve the system for the scalar coefficients using substitution or Gaussian elimination.

A linear combination is unique when the basis vectors are linearly independent and span the entire space. For n linearly independent vectors in ℝⁿ, the representation of any target vector is unique.

If the system of equations has no solution, the target vector does not lie in the span of the given basis vectors. This means the basis vectors do not span the full space containing the target vector.

The span of a set of vectors is the set of ALL possible linear combinations of those vectors. A single linear combination is one specific result; the span is the collection of every possible result.

Yes, if the basis vectors are linearly dependent or if there are more basis vectors than the dimension of the space, there may be infinitely many valid sets of coefficients that produce the same target vector.

The zero vector can always be expressed as the trivial linear combination where all coefficients are zero. If the only way to express the zero vector as a linear combination is with all zero coefficients, the vectors are linearly independent.

Negative coefficients mean the corresponding vector is subtracted rather than added. For example, 2v₁ - 3v₂ means adding twice v₁ and subtracting three times v₂.

A convex combination is a special linear combination where all coefficients are non-negative and sum to 1. The result always lies within the convex hull of the given vectors.

Set up the system [v₁ v₂ ... vₙ]x = v_target and solve. If a solution exists, the vector is in the span. If the system is inconsistent, it is not.

An affine combination requires the coefficients to sum to 1 but allows negative values. A convex combination additionally requires all coefficients to be non-negative.

Yes, linear combinations can use complex scalar coefficients. This is common in quantum mechanics and signal processing where vectors exist in complex vector spaces.

Functions can be combined the same way as vectors: f(x) = c₁f₁(x) + c₂f₂(x) + ... + cₙfₙ(x). This is fundamental in Fourier analysis, polynomial approximation, and functional analysis.

Form a matrix A with the basis vectors as columns, then solve the system Ax = b using Gaussian elimination or matrix inversion: x = A⁻¹b when A is square and invertible.

A set of vectors is called a generating set (or spanning set) for a vector space if every vector in the space can be written as a linear combination of vectors from the set.

A basis is a set of vectors that is both linearly independent and spans the entire space. Any vector in the space can be uniquely expressed as a linear combination of basis vectors.

The coordinate vector of v with respect to a basis B is the vector [v]_B = (c₁, c₂, ..., cₙ) where c₁, c₂, ..., cₙ are the coefficients of the linear combination expressing v in terms of the basis B.

Yes, the calculator works for any dimension. For 3D vectors, you set up three equations (one per component) and solve for the coefficients. With three linearly independent 3D vectors, any 3D target can be expressed.

When there are more basis vectors than dimensions, the system is underdetermined and may have infinitely many solutions (or no solution). The basis vectors are necessarily linearly dependent in this case.

In computer graphics, linear combinations are used for transforming coordinates, blending animations, interpolating colors, and expressing points in different coordinate systems (world space, screen space, etc.).

The Linear Combination Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Linear Combination Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.