alleryStart learning →For an $m \times n$ matrix $A$, its RANK equals
A$m$, the row count
B$n$, the column count
Cdim column space = dim row space
D$m + n$, sum of dimensions
Answer & Solution
Correct answer: C. dim column space = dim row space
1. RANK of $A$ = dimension of its COLUMN SPACE (the span of its columns in $\mathbb{R}^m$).
2. ROW RANK = COLUMN RANK theorem: dimension of row space = dimension of column space.
3. Both equal the number of pivots in the row-reduced echelon form.
4. Bound: $\text{rank}(A) \leq \min(m, n)$.
5. Options A, B, D give wrong upper bounds or sums.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §3.5 (Row rank = column rank)._
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