Enter Matrix Data
Example Data Table
| Matrix Input | Target Vector | Expected Idea | Main Result |
|---|---|---|---|
1,0,1 / 0,1,1 / 1,1,2 |
2,3,5 |
Columns act as vectors in R³. | Rank is 2, so the full R³ is not spanned. |
1,0,0 / 0,1,0 / 0,0,1 |
4,5,6 |
Standard basis vectors are used. | Rank is 3, so R³ is spanned. |
1,2 / 3,6 |
5,15 |
The second column is a multiple. | The set is dependent and spans a line. |
Formula Used
Let the matrix be A = [v1 v2 ... vn], where each column is a vector.
span(v1, v2, ..., vn)is the set of all combinationsc1v1 + c2v2 + ... + cnvn.rank(A)is the number of pivot columns in the reduced row echelon form.- The vectors span
R^mwhenrank(A) = m. - The vectors are independent when
rank(A) = n. - A target vector
bis in the span whenrank(A) = rank([A|b]). nullity(A) = n - rank(A), which shows the number of free variables.
How to Use This Calculator
- Enter the matrix values in the large input box.
- Select whether each vector is stored as a row or column.
- Add a target vector if you want a membership test.
- Adjust tolerance if your data contains tiny decimal noise.
- Press the calculate button.
- Review rank, pivots, basis vectors, RREF, and graph output.
- Use CSV or PDF buttons to save the result.
Spanning Set Matrix Guide
What a Spanning Set Means
A spanning set is a group of vectors that can build every vector in a chosen space. The building process uses linear combinations. Each vector receives a scalar coefficient. The calculator places the vectors into a matrix. Then it studies the pivots. Pivots reveal the real coverage of the vector set.
Why Rank Matters
Rank is the key measurement. It counts the number of independent directions. If vectors live in R³, three independent directions are needed. If rank is only two, the vectors cover a plane. If rank is one, they cover a line. If rank is zero, every vector is zero.
Using Row Reduction
Row reduction changes the matrix without changing its column relationships. The reduced form shows pivot columns clearly. These pivot columns form a basis for the column space. Non-pivot columns are dependent on earlier pivot columns. This makes the method useful for proofs and homework checks.
Target Vector Testing
Many problems ask whether a vector belongs to a span. This calculator augments the matrix with the target vector. Then it compares ranks. Equal ranks mean the target is reachable. Unequal ranks mean no coefficient set can create it. When possible, the tool also gives one coefficient solution.
Practical Learning Value
The result helps students see more than one answer. It shows rank, nullity, pivots, determinant, basis vectors, and graphs. Two-dimensional and three-dimensional inputs are plotted as vectors. Higher-dimensional inputs use vector norms. This gives a clean visual clue. It also makes the calculator useful for quick review.
FAQs
1. What is a spanning set?
A spanning set is a group of vectors whose linear combinations can create every vector in a target space or subspace.
2. How does this calculator test spanning?
It forms a matrix from the vectors, reduces it to echelon form, counts pivots, and compares rank with the vector dimension.
3. What does rank mean here?
Rank is the number of independent directions in the matrix. It equals the number of pivot columns after row reduction.
4. When do vectors span Rn?
Vectors span Rn when the matrix rank equals n. For R³, the rank must be three.
5. What is a pivot column?
A pivot column contains a leading one in reduced row echelon form. Pivot columns identify basis vectors.
6. Can dependent vectors still span a space?
Yes. A set can contain extra dependent vectors and still span the full space if enough independent directions exist.
7. What does target vector membership mean?
It checks whether the target vector can be written as a linear combination of the entered vectors.
8. Why use a tolerance value?
Small decimal errors can appear during row operations. Tolerance treats tiny values as zero for cleaner numerical results.