Calculator Form
Each line is one vector. Separate entries with commas or spaces.
Example Data Table
| Example | Candidate vectors | Target vector | Expected rank | Spans ambient space? | Target in span? |
|---|---|---|---|---|---|
| Full span in R³ | [1,0,1], [0,1,1], [1,1,0] | [2,3,1] | 3 | Yes | Yes |
| Dependent set in R³ | [1,0,0], [0,1,0], [1,1,0] | [0,0,1] | 2 | No | No |
| Spanning pair in R² | [1,2], [3,1] | [5,5] | 2 | Yes | Yes |
Formula Used
1) Column matrix construction
If the entered vectors are v₁, v₂, ..., vm in Rn, the calculator forms A = [v₁ v₂ ... vm]. The columns of A are exactly the candidate spanning vectors.
2) Rank test for spanning
The set spans Rn exactly when rank(A) = n. After Gaussian elimination, this means every row receives a pivot. If at least one row has no pivot, the set misses some direction in the space.
3) Target membership test
A target vector b lies in span{v₁,...,vm} exactly when rank(A) = rank([A|b]). If the augmented matrix produces a contradiction row, such as 0 = 1, the target is outside the span.
4) Basis subset rule
The pivot columns of the original matrix A form a basis for the column space. Those original vectors give a minimal spanning subset of all entered vectors.
5) Square case shortcut
When A is square, det(A) ≠ 0 gives the same conclusion as full rank: the entered vectors are linearly independent and span the whole ambient space.
How to Use This Calculator
Step 1: Enter one vector per line in the candidate box.
You can separate entries with commas, spaces, or semicolons.
Step 2: Add a target vector if you want a membership proof.
Leave it blank when you only need a spanning-set result.
Step 3: Set tolerance and display precision.
Tolerance controls when tiny floating values are treated as zero.
Step 4: Press Analyze Spanning Set.
The result appears above the form, with rank, pivots, basis vectors, coefficients, proof notes, and row-reduction steps.
Step 5: Export your work.
Use the CSV button for a compact table or the PDF button for a clean summary report.
Frequently Asked Questions
1) What does it mean for vectors to span a space?
A set spans a space when every vector in that space can be written as a linear combination of the set. In matrix language, the columns generate every reachable direction in the ambient space.
2) Why does rank decide the spanning result?
Rank counts pivot directions. If rank equals the ambient dimension, the columns cover every independent direction required for the whole space. If rank is smaller, at least one direction is missing.
3) What are pivot columns used for?
Pivot columns mark the original vectors that remain essential after row reduction. Those vectors form a basis for the column space and create a minimal spanning subset.
4) Can a spanning set still be linearly dependent?
Yes. A set may span the space and still contain redundant vectors. That happens when some vectors can already be built from the others, even though the whole collection still reaches every direction.
5) Why is the target vector optional?
Spanning and target membership are related but different questions. You may only need to know whether the set spans the whole ambient space, without testing one specific target vector.
6) Why can coefficient solutions be nonunique?
If the spanning set is dependent, some columns are free. Then different coefficient choices can produce the same target vector. The calculator shows one valid basic solution by setting free variables to zero.
7) When does the determinant help?
The determinant only applies when the matrix is square. A nonzero determinant means the vectors are independent and span the full space. A zero determinant means the square set fails that test.
8) What graph appears for dimensions above three?
Higher-dimensional vectors cannot be drawn directly as arrows on a standard screen, so the tool switches to a Plotly comparison view using vector norms and any available coefficient values.