Row space: R(A) = C(Aᵀ) = span of rows of A. Dimension = rank. Row operations preserve row space, so non-zero rows of REF form a basis. R(A) ⊥ N(A) (orthogonal complement in Rⁿ). dim R(A) + dim N(A) = n.
How to Use
Enter a matrix:
- A: Any m×n matrix
- Output: Row space basis
- Rank: Dimension
Row vs Column Space
Same dimension (rank), but different spaces: rows ⊂ Rⁿ, columns ⊂ Rᵐ. Row operations preserve row space but change column space. Column operations preserve column space but change row space.
Orthogonal Complement
Row space ⊥ null space in Rⁿ. Every vector in Rⁿ = row space component + null space component. This is the fundamental theorem of linear algebra (part 2).
Step-by-Step Instructions
- 1Enter a matrix.
- 2Row reduce.
- 3Take non-zero rows.
- 4Get basis.
- 5Check dimension.