The RANK-NULLITY THEOREM states that for $T : V \to W$ on a finite-dim $V$
A$\dim(\ker T) = \dim(\text{im}\,T)$
B$\dim(\ker T) \cdot \dim(\text{im}\,T) = \dim V$
C$\dim V = \dim W$
D$\dim(\ker T) + \dim(\text{im}\,T) = \dim V$
Answer & Solution
Correct answer: D. $\dim(\ker T) + \dim(\text{im}\,T) = \dim V$
1. RANK-NULLITY: for a linear map $T : V \to W$ on a finite-dim $V$:
$\dim V = \dim(\ker T) + \dim(\text{im}\,T)$.
2. NULLITY = $\dim(\ker T)$; RANK = $\dim(\text{im}\,T)$.
3. Consequence: if $\dim V = n$ and rank $= r$, then nullity $= n - r$.
4. Useful for solving $Ax = 0$ (null space) and counting equations.
5. Options A, B, C are incorrect statements.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §3.3 (Rank-Nullity theorem)._
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