Systems of Equations Calculator
Solve 2×2 and 3×3 linear systems using Cramer's Rule, elimination, substitution, or matrix methods. Shows solution, determinants, verification, and method comparison.
Linear Combination Calculator
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.
Target Vector
Basis Vector 1
Basis Vector 2
Linear Combination⧉
3.3333·v1 + 0.3333·v2
t = c₁v₁ + c₂v₂
Coefficient c1⧉
3.333333
Scalar multiplier for basis vector v1 = (1, 2)
Coefficient c2⧉
0.333333
Scalar multiplier for basis vector v2 = (2, 1)
Determinant⧉
-3.000000
Determinant of the basis matrix — nonzero means a unique solution exists
Verification⧉
(4.000000, 7.000000)
Should equal target (4, 7)
Target Magnitude⧉
8.062258
‖t‖ — Euclidean length of the target vector
Vector Component Breakdown
x = 4.00
3.3
y = 7.00
6.7
c1·v1c2·v2
Coefficient Breakdown
| Component | c1 · v1 | c2 · v2 | Sum | Target |
|---|---|---|---|---|
| x | 3.3333 | 0.6667 | 4.0000 | 4.0000 |
| y | 6.6667 | 0.3333 | 7.0000 | 7.0000 |
Basis Vectors Summary
| Vector | Components | Magnitude | Coefficient | Contribution ‖c·v‖ |
|---|---|---|---|---|
| v1 | (1, 2) | 2.2361 | 3.3333 | 7.4536 |
| v2 | (2, 1) | 2.2361 | 0.3333 | 0.7454 |
| Target | (4, 7) | 8.0623 | — | — |
Planning notes, formulas, and examples
About the Linear Combination Calculator
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, while manual entry handles any vectors you need.
When This Page Helps
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.
How to Use the Inputs
- Enter x and y in the input fields.
- Select the mode, method, or precision options that match your linear combination problem.
- Read Linear Combination first, then use Determinant to confirm your setup is correct.
- Try a preset such as "(3,5) = a(1,0)+b(0,1)" to test a known case quickly.
Formula used
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.Example Calculation
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.
Tips & Best Practices
- If the determinant is zero, the basis vectors are linearly dependent and may not span the target.
- In 2D, two non-parallel vectors always form a basis for the plane.
- Larger absolute determinant means the basis vectors are "more independent" — less numerical sensitivity.
- For orthogonal bases, the coefficients simplify to dot-product projections: cᵢ = (t · vᵢ) / (vᵢ · vᵢ).
- Use this calculator to check change-of-basis problems in homework or exams quickly.
How This Linear Combination Calculator Works
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.
Interpreting Results
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.
Study Strategy
Sources & Methodology
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Frequently Asked Questions
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.
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.
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).