Row Space Calculator
Calculate row space, column space bases, rank, and the four fundamental subspaces.
Matrix Input
Matrix A
Rank of Matrix
2
Row Space
dim = 2
Column Space
dim = 2
Null Space
dim = 1
Left Null Space
dim = 1
Row Space Basis
r1 = (1.0000, 0.0000, -1.0000)
r2 = (0.0000, 1.0000, 2.0000)
Column Space Basis
Pivot columns: 1, 2
c1 = (1.0000, 4.0000, 7.0000)
c2 = (2.0000, 5.0000, 8.0000)
RREF
1.000.00-1.00
0.001.002.00
0.000.000.00
Four Fundamental Subspaces
Row Space & Null Space
Row space is in Rⁿ, null space is in Rⁿ. They are orthogonal complements: dim(row) + dim(null) = n.
Column Space & Left Null Space
Column space is in Rᵐ, left null space is in Rᵐ. They are orthogonal complements: dim(col) + dim(left null) = m.