Calculator Input
Enter a matrix with spaces or commas between numbers and one row per line. Choose whether you want the orthogonal complement of the row space or column space.
Example Data Table
| Example | Matrix | Target | Rank | Complement Dimension | Basis |
|---|---|---|---|---|---|
| Example 1 | [1 2 3; 2 4 6] | Row space complement | 1 | 2 | {(-2, 1, 0), (-3, 0, 1)} |
| Example 2 | [1 0; 0 1] | Column space complement | 2 | 0 | Trivial complement only |
Formula Used
For a matrix A, the orthogonal complement of the row space is the null space of A. That means every vector x in the complement satisfies Ax = 0.
For the column space, the orthogonal complement is the null space of AT. Every vector y in that complement satisfies ATy = 0.
The calculator uses reduced row echelon form to identify pivot columns and free variables. Each free variable produces one basis vector for the orthogonal complement.
Dimension rule: dim(S⊥) = n − dim(S) inside an ambient space of dimension n. This follows directly from the rank-nullity theorem.
How to Use This Calculator
- Type the matrix entries in the textarea, placing one row on each line.
- Choose whether you want the orthogonal complement of the row space or the column space.
- Set the display precision and numeric tolerance if your matrix contains decimals or near-zero values.
- Press the calculate button to see the result block under the header and above the form.
- Review the rank, complement dimension, RREF matrix, basis vectors, and residual checks.
- Use the CSV or PDF buttons to save your current output for reporting, study notes, or audits.
FAQs
1. What does an orthogonal complement represent?
It is the set of all vectors perpendicular to every vector in a chosen subspace. For matrices, it is usually obtained by solving a homogeneous linear system.
2. Why is the row-space complement found from Ax = 0?
Each row of A acts like a dot product constraint. Solving Ax = 0 finds every vector whose dot product with every row is zero.
3. Why is the column-space complement found from Aᵀy = 0?
A vector orthogonal to every column must have zero dot product with each column. Writing those dot products together produces Aᵀy = 0.
4. What do pivot and free columns tell me?
Pivot columns determine the rank and the dependent variables. Free columns create the independent directions that become basis vectors of the orthogonal complement.
5. What happens when the complement dimension is zero?
The orthogonal complement is trivial. That means the only vector in it is the zero vector, so no nonzero basis vectors are listed.
6. Can I use decimal entries?
Yes. The tolerance field helps treat tiny rounding artifacts as zeros, which is especially useful when your matrix comes from measured or computed data.
7. Why does the calculator show residual checks?
Residual vectors verify the basis numerically. A correct basis should produce zero or near-zero residuals when substituted into the relevant homogeneous system.
8. When should I choose row space instead of column space?
Choose row space when vectors live in the matrix column count. Choose column space when vectors live in the matrix row count.