Calculator Input
Example Data Table
| Row | v1 | v2 | v3 | b |
|---|---|---|---|---|
| 1 | 1 | 0 | 2 | 3 |
| 2 | 2 | 1 | 1 | 5 |
| 3 | -1 | 3 | 0 | 5 |
| Exact Coefficients | 1 | 2 | 1 | — |
Formula Used
Main equation: A · c = b
Matrix form: A = [v1 v2 ... vn], where each column is a vector.
Consistency test: rank(A) = rank([A|b])
Residual: r = b - A · c
Approximation when exact consistency fails: (AᵀA)c = Aᵀb
A target vector b is a linear combination of the columns of A when there exists a coefficient vector c such that multiplying A by c reproduces b exactly.
The calculator forms the augmented matrix [A|b], reduces it to RREF, and uses the rank test to decide whether the system is consistent.
If the exact system is inconsistent, the tool still reports the closest least-squares approximation, along with the residual vector and residual norm.
How to Use This Calculator
- Select the vector dimension and the number of column vectors.
- Enter the entries for each column of matrix A.
- Enter the entries for the target vector b.
- Click Calculate Linear Combination.
- Review the classification, coefficients, RREF, reconstruction, residual, and graph.
- Use the CSV or PDF buttons to download your result summary.
Answers to Common Search Questions
determine if b is a linear combination of the vectors formed from the columns of the matrix a
Place the vectors as columns of A, solve A·c = b, and inspect consistency. If the augmented matrix is consistent, b lies in the column space of A and is a linear combination. This calculator performs that test automatically and returns coefficients, rank checks, and residuals.
What is a linear combination of vectors?
A linear combination is a weighted sum of vectors. If you multiply each vector by a scalar coefficient and add the results, the new vector is a linear combination of the original vectors. In matrix form, that idea becomes A·c = b.
linear combination of vectors calculator
This calculator checks whether a target vector can be built from the columns of a matrix. It reports exact coefficients when possible, shows least-squares output when not, displays RREF, provides a Plotly graph for 2D or 3D data, and exports results to CSV or PDF.
determine if b is a linear combination of the vectors formed from the columns of the matrix a.
Use the rank condition rank(A) = rank([A|b]). When the ranks match, the system is consistent and b is inside the span of A’s columns. When the ranks differ, no exact linear combination exists, so the calculator shows the closest approximation and residual instead.
FAQs
1. Can this calculator tell whether b is in the span of A?
Yes. It forms the augmented matrix [A|b], computes RREF, compares ranks, and then states whether b is an exact linear combination of A’s columns.
2. What happens when there are infinitely many solutions?
The tool labels the system as an exact solution with free variables. It then reports one valid coefficient set by assigning zero to free variables.
3. What if no exact combination exists?
The calculator reports that b is not an exact linear combination and then shows a least-squares approximation, reconstructed vector, residual vector, and residual norm.
4. Why is RREF useful here?
RREF reveals pivots, free variables, and inconsistent rows. That makes it ideal for checking span membership and reading coefficient relationships clearly.
5. Do dependent columns still work?
Yes. Dependent columns are allowed. They may create free variables or multiple valid combinations, but the calculator still checks consistency correctly.
6. Does column order matter?
Yes. The order changes which coefficient belongs to which vector, although the span itself stays the same if the same columns are present.
7. When is the graph shown?
A Plotly graph appears for two-dimensional and three-dimensional inputs. Four-dimensional cases are still solved, but they are not plotted visually.
8. What sizes are supported in this version?
This page supports dimensions from 2 to 4 and from 2 to 5 column vectors, which covers many classroom and homework examples.