Calculator
Example Data Table
| Sample Matrix | Span Type | Basis | Rank | Notes |
|---|---|---|---|---|
| [1 2 3] [2 4 6] [1 1 2] |
Column span | {<1, 2, 1>, <2, 4, 1>} | 2 | Column 3 equals Column 1 plus Column 2. |
| [1 0 2 1] [0 1 1 2] [2 1 5 4] |
Row span | Nonzero rows of RREF | 2 | The third row depends on the first two rows. |
Formula Used
The span of selected vectors contains every linear combination of those vectors. For a set v1, v2, ..., vk, the span is all vectors c1v1 + c2v2 + ... + ckvk.
To find a basis efficiently, the calculator reduces the matrix to reduced row echelon form. The number of pivot positions equals the rank, and the rank is the dimension of the span.
For column span, pivot columns from the original matrix form the basis. For row span, the nonzero rows of reduced row echelon form form the basis.
How to Use This Calculator
- Enter the matrix with one row per line.
- Separate entries using spaces, commas, or semicolons.
- Choose column span or row span.
- Select the decimal precision and pivot tolerance.
- Click Compute Span to see rank, basis vectors, pivots, and the graph.
- Use the CSV or PDF buttons to export the results.
Frequently Asked Questions
1. What does the span of a matrix mean?
It describes every vector you can build from linear combinations of the chosen rows or columns. The calculator lets you study either the column space or the row space.
2. Why are pivot columns important?
Pivot columns identify which original columns are linearly independent. Those columns form a basis for the column space and remove redundant vectors from the spanning set.
3. Why does the calculator use reduced row echelon form?
Reduced row echelon form reveals pivot positions clearly. It makes the rank easy to read and gives a clean way to extract basis rows for the row space.
4. What is the difference between rank and nullity?
Rank counts independent columns or rows. Nullity counts free variables in the associated homogeneous system. Together they satisfy rank plus nullity equals the number of columns.
5. Can I enter fractions instead of decimals?
Yes. Entries such as 1/2, -3/4, and 5 are accepted. Keep each fraction inside one cell entry and separate entries with spaces, commas, or semicolons.
6. Why might the graph not appear?
The graph is only shown when the basis vectors live in two or three dimensions. Higher-dimensional spaces still compute correctly, but they cannot be displayed directly on a simple plot.
7. What happens for the zero matrix?
The zero matrix has rank zero. Its row space and column space both equal the zero subspace, so there are no nonzero basis vectors to list.
8. Can this help with homework checking?
Yes. It quickly verifies basis choices, pivot columns, dependence, rank, and reduced row echelon form. The exports are useful when you want a saved record of the solution.