Result
Advanced Vector Linear Combination Calculator
Enter source vectors, coefficients, and an optional target vector. The calculator can compute a direct combination or solve whether a target is in the span.
Example Data Table
This example uses three vectors in R³ and coefficients 2, -1, and 3.
| Vector | Components | Coefficient | Scaled Vector |
|---|---|---|---|
| v₁ | (1, 2, 3) | 2 | (2, 4, 6) |
| v₂ | (0, -1, 4) | -1 | (0, 1, -4) |
| v₃ | (2, 1, 0) | 3 | (6, 3, 0) |
| Sum | 2v₁ - v₂ + 3v₃ | (8, 8, 2) | |
Formula Used
A linear combination of vectors uses scalars multiplied by source vectors:
c₁v₁ + c₂v₂ + ... + cₖvₖ = b
For source vectors placed as columns of a matrix, the same idea becomes:
A c = b
Here, A is the vector matrix, c is the coefficient vector, and b is the target vector. The calculator uses row reduction to test if b is in the span of the source vectors. It also reports rank, pivot columns, free variables, residuals, and coefficient solutions when possible.
How to Use This Calculator
- Choose the operation mode.
- Enter the vector dimension.
- Enter the number of source vectors.
- Type one source vector per line.
- Enter coefficients for a direct combination.
- Add a target vector if you want a span test.
- Choose decimal precision and tolerance.
- Press calculate to view the result above the form.
- Use CSV or PDF export for saving your work.
Linear Combination of Vectors Guide
What It Means
Vectors appear in many math topics. They describe movement, forces, data, and coordinates. A linear combination builds one vector from other vectors. Each source vector is multiplied by a scalar. The scaled vectors are then added component by component. This idea looks simple, yet it supports span, basis, rank, systems, and transformations.
Why This Calculator Helps
This calculator helps you test that idea with clear steps. You can enter any small or medium vector set. You can also enter coefficients to compute a direct combination. When a target vector is included, the tool compares the result with that target. The residual tells you how far the computed vector is from the target. A zero residual means the coefficients match the target within the selected tolerance.
Solving Target Vectors
The solve option uses the vectors as columns of a matrix. It then solves the matrix equation A c = b. Here A contains the source vectors. The unknown vector c contains the needed coefficients. The target vector is b. Row reduction shows whether a solution exists. If the augmented matrix has a contradiction, the target is not in the span. If a solution exists, the target belongs to the span.
Rank and Dependence
Advanced users can check rank and pivot columns. A unique solution usually appears when every coefficient variable has a pivot. Infinite solutions can appear when extra vectors are present. That case means the same target can be built in more than one way. This often happens when vectors are dependent.
Best Use
Use this page for homework, lessons, and quick verification. Keep vector dimensions consistent. Enter one vector per line. Use commas, spaces, or tabs between components. Start with the example table if you are learning the format. Then change the values slowly. Download results when you need a record. The exported files can support notes, lab work, tutorials, and reports. Always review the displayed steps. They help reveal entry errors. They also make the calculation easier to explain.
Accuracy Tip
For best accuracy, choose a tolerance that fits your class rules. Smaller tolerances are stricter. Larger tolerances are helpful for decimal data. The calculator does not replace reasoning. It organizes arithmetic so you can focus on the meaning of the result clearly.
FAQs
What is a linear combination of vectors?
A linear combination is a sum of scaled vectors. Each vector is multiplied by a coefficient. The scaled vectors are then added component by component to form a new vector.
What does span mean?
Span is the set of all vectors that can be created from given source vectors using linear combinations. If a target vector can be built this way, it is in the span.
How should I enter vectors?
Enter one vector on each line. Separate components with commas, spaces, or tabs. Each vector must have the same number of components as the selected dimension.
What are coefficients?
Coefficients are scalar values that multiply each source vector. For example, in 2v₁ - 3v₂, the coefficients are 2 and -3.
What does residual mean?
The residual is the difference between the computed vector and the target vector. A zero or very small residual means the computed vector matches the target closely.
What is row reduction used for?
Row reduction solves the matrix equation A c = b. It shows whether the target vector can be created from the source vectors.
Why can there be infinite solutions?
Infinite solutions happen when vectors are dependent or extra vectors are available. Then a target vector may be created by more than one coefficient set.
Can I download the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean printable report with inputs, results, and calculation notes.