Last updated: August 1, 2026
Linear Combination Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Enter one generating vector per row and a target vector of matching dimension. The calculator solves the corresponding linear system for the coefficient vector and then rebuilds the target from those coefficients so you can verify the linear combination directly.
Enter one generator per row and a target vector to solve for the coefficients that reproduce the target as a linear combination.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Solve Vᵀc = b, where the rows of V are the vectors being combined and b is the target vector.
Where:
- V=matrix of input vectors
- c=coefficient vector
- b=target vector
Worked Examples
Standard basis in R²
Express [3,4] in the standard basis.
- 1Build the system from the basis rows.
- 2Solve for the coefficient vector.
- 3Reconstruct the target to verify it.
Non-orthogonal basis
Use [1,1] and [1,-1] to build [4,2].
- 1Set c₁[1,1] + c₂[1,-1] = [4,2].
- 2Solve the 2×2 system.
Three vectors in R³
Use the identity basis for [5,7,9].
- 1Identity basis leaves the target unchanged.
- 2Read the coefficients directly.
Singular vector set
Rows [1,2] and [2,4] do not give a unique solution.
- 1Detect the dependent rows.
- 2Reject the non-invertible system.
Introduction
Solve for the coefficients that express a target vector as a combination of a square set of vectors.
Overview
Solve for the coefficients that express a target vector as a combination of a square set of vectors.
Coefficient vector
Span checks
Coordinate changes
Basis practice problems
Input format
Enter one vector per row, such as 1,1;1,-1.
Separate rows with semicolons
Separate entries with commas or spaces
Brackets are optional
Keep row lengths consistent
Formula and method
Solve Vᵀc = b, where the rows of V are the vectors being combined and b is the target vector.
- V matters:
matrix of input vectors.
- C matters:
coefficient vector.
- B matters:
target vector.
Validation rules
The calculator uses deterministic validation before it computes the final result.
The number of input vectors must equal the vector dimension
The target length must match the vector dimension
Dependent generating sets are rejected
Every entry must be finite
Interpreting the outputs
Use the main result first, then the secondary fields to verify the structure and meaning of the answer.
A successful result means the target lies in the span of the rows
The coefficients are listed in the same order as the input vectors
A zero coefficient means that vector does not contribute
The reconstructed vector is a direct check
Common use cases
These are typical reasons to use this calculator in study or applied work.
Span checks
Coordinate changes
Basis practice problems
Introductory system solving
FAQs
How do I enter the vectors?
Enter one vector per row, with coordinates separated by commas or spaces.
Why must the system be square?
This calculator is designed for a unique coefficient vector, so the number of vectors must match the dimension.
What if my vectors are dependent?
Dependent rows make the system singular, so the calculator returns an error.
Does the vector order matter?
Yes. Reordering the input rows reorders the coefficients.
Can I use decimals?
Yes. Any finite real values are supported.
What does reconstructedVector show?
It shows the target rebuilt from the reported coefficients.