Express a target vector as a linear combination of basis vectors in 2D or 3D. Find scalar coefficients, verify the solution, and visualise component contributions.
A linear combination expresses a vector as a sum of scalar multiples of other vectors: t = c₁v₁ + c₂v₂ + … + cₙvₙ. This calculator finds the scalar coefficients c₁, c₂, and (in 3D) c₃ that combine your chosen basis vectors to produce a given target vector. It supports both 2D and 3D modes and verifies the solution by reconstructing the target from the computed coefficients.
Linear combinations are a cornerstone of linear algebra, appearing in nearly every topic from systems of equations to change-of-basis transformations, span and independence checks, and coordinate representations. In applied fields, they underlie computer graphics transformations, signal processing, machine learning feature engineering, and physics force decomposition.
The calculator computes the determinant of the basis matrix (a nonzero determinant guarantees a unique solution), displays each coefficient, and breaks down the contribution of each basis vector to every component of the target in a colour-coded bar chart and summary table. Eight presets for both 2D and 3D let you explore classic examples instantly, while manual entry handles any vectors you need.
Linear Combination Calculator helps you solve linear combination problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter x, y, z once and immediately inspect Linear Combination, Determinant, Verification to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Solve the matrix equation [v₁ | v₂ | … | vₙ]·c = t using Cramer's rule. For 2D: c₁ = (t₁v₂₂ − t₂v₂₁)/det, c₂ = (v₁₁t₂ − v₁₂t₁)/det, where det = v₁₁v₂₂ − v₁₂v₂₁.
Result: Linear Combination shown by the calculator
Using the preset "(3,5) = a(1,0)+b(0,1)", the calculator evaluates the linear combination setup, applies the selected algebra rules, and reports Linear Combination with supporting checks so you can verify each transformation.
This calculator takes x, y, z and applies the relevant linear combination relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Linear Combination, Determinant, Verification, Target Magnitude to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
A linear combination of vectors v₁, v₂, …, vₙ is any expression c₁v₁ + c₂v₂ + … + cₙvₙ where c₁ through cₙ are scalars. It is the most basic operation in linear algebra.
A unique set of coefficients exists when the basis vectors are linearly independent, which happens if and only if the determinant of the basis matrix is nonzero. Use this as a practical reminder before finalizing the result.
If the determinant is zero, either no solution exists (the target is not in the span) or infinitely many solutions exist. This calculator will warn you about the zero determinant.
This calculator handles the standard square case (2 vectors in 2D, 3 in 3D). Over-determined systems require least-squares methods, which involve a different algorithm.
Finding a linear combination is equivalent to solving a system of linear equations Ax = b, where A is the matrix of basis columns and b is the target vector. Keep this note short and outcome-focused for reuse.
In 2D, |det| is the area of the parallelogram formed by the basis vectors. In 3D, |det| is the volume of the parallelepiped. A zero determinant means the vectors are co-planar (3D) or co-linear (2D).