Span of Vectors Calculator

Example Data Table

Formula Used

The span of vectors v1, v2, ..., vk is the set of all linear combinations.

Span(v1, v2, ..., vk) = c1v1 + c2v2 + ... + ckvk

The coefficients c1, c2, ..., ck are real numbers. The calculator builds a matrix A with the input vectors as columns.

Dimension of span = rank(A)

A target vector b belongs to the span when the system Ac = b is consistent.

b is in Span(A) when rank(A) = rank([A | b])

How to Use This Calculator

1. Enter one vector per line in the vectors box.
2. Separate vector components with commas or spaces.
3. Enter a target vector when you want membership testing.
4. Choose the analysis mode and decimal precision.
5. Press the calculate button.
6. Review rank, pivot columns, basis vectors, and RREF output.
7. Use CSV or PDF export for records.

Understanding Vector Span

A vector span describes every vector you can build from a given list. Each input vector acts like a direction. Each scalar coefficient stretches that direction. When you add those scaled directions, you create a linear combination. The set of all possible linear combinations is the span.

Why Span Matters

Span is central in algebra, geometry, data science, and engineering. It shows whether vectors cover a line, a plane, or a higher dimensional space. It also explains redundancy. A vector is redundant when other vectors already create the same direction. Removing redundant vectors can simplify a model without losing coverage.

Rank and Basis

The calculator places your vectors into a matrix. Each vector becomes a column. Row reduction then finds pivot columns. Pivot columns identify a basis for the span. The number of pivot columns is the rank. Rank is also the dimension of the span. If rank equals the ambient dimension, the vectors span the whole space. If rank is smaller, they span only a subspace.

Target Membership

You may enter a target vector. The calculator checks whether the target can be written as a linear combination of the input vectors. It compares the rank of the coefficient matrix with the rank of the augmented matrix. Equal ranks mean the target belongs to the span. Different ranks mean it does not belong.

Interpreting Results

A rank of one means all nonzero vectors point along one line. A rank of two usually means a plane inside three dimensional space. A rank of three in three dimensional space means full coverage. Extra vectors can still be useful, but they may not add new directions. The basis list shows the smallest detected set of original vectors that keeps the same span.

Good Input Practice

Use one vector per line. Keep every vector the same length. Separate components with commas or spaces. Use decimal values when needed. Set a small tolerance for noisy data. A larger tolerance may treat tiny values as zero. A smaller tolerance keeps more numerical detail. Review the reduced matrix when results look surprising.

Use the export buttons after calculation. Save the summary for homework, reports, tutoring notes, or quick later comparison and accurate classroom review.

FAQs