Column space: C(A) = {Ax : x ∈ Rⁿ} = span of columns of A. Dimension = rank. Basis = pivot columns of original A (not REF). b ∈ C(A) ⟺ Ax = b is consistent. For m×n matrix: C(A) ⊂ Rᵐ.
How to Use
Enter a matrix:
- A: Any m×n matrix
- Output: Column space basis
- Rank: Dimension
Finding Basis
Row reduce A. Pivot columns in REF correspond to basis vectors — but use the ORIGINAL columns of A (not the reduced ones). This is an important subtlety!
Four Subspaces
- C(A): column space ⊂ Rᵐ, dim = r
- N(A): null space ⊂ Rⁿ, dim = n−r
- C(Aᵀ): row space ⊂ Rⁿ, dim = r
- N(Aᵀ): left null space ⊂ Rᵐ, dim = m−r
Step-by-Step Instructions
- 1Enter a matrix.
- 2Row reduce.
- 3Identify pivot columns.
- 4Get basis from originals.
- 5Check dimension.