According to the Rank-Nullity Theorem, for an $m \times n$ real matrix $A$, the dimension of the null space $\text{nullity}(A)$ and the rank $\text{rank}(A)$ satisfy:
Options:
(A) $\text{rank}(A) + \text{nullity}(A) = n$
(B) $\text{rank}(A) + \text{nullity}(A) = m$
(C) $\text{rank}(A) - \text{nullity}(A) = n$
(D) $\text{rank}(A) \times \text{nullity}(A) = m n$
Answer Key:
A
Step-by-Step Solution & Explanation
The Rank-Nullity Theorem states that for any linear map represented by an $m \times n$ matrix $A$, $\text{dim}(\text{col}(A)) + \text{dim}(\text{null}(A)) = n$ (the number of columns).