Enter matrix dimensions below, generate the input matrix, fill in the values or generate a random integer matrix, and then calculate the bases for the column space, nullspace, row space, and left nullspace of the matrix $A$.
For an $m \times n$ matrix $A$:
Rank-Nullity Theorem:
$$ \text{rank}(A) + \text{nullity}(A) = n \quad \text{(number of columns)} $$
This states that the dimension of the column space plus the dimension of the nullspace equals the number of columns.
Row Rank = Column Rank:
$$ \text{rank}(A) = \text{rank}(A^T) $$
The dimension of the column space $C(A)$ equals the dimension of the row space $C(A^T)$.
Orthogonal Complements:
$$ C(A) \oplus N(A^T) = \mathbb{R}^m \quad \text{and} \quad C(A^T) \oplus N(A) = \mathbb{R}^n $$
The column space and left nullspace are orthogonal complements in $\mathbb{R}^m$. The row space and nullspace are orthogonal complements in $\mathbb{R}^n$.