Question

Let U and V be vector spaces over a field F and dim U = n. Let T:U→V
be a linear operator, then rank (T) + nullity (T) = ...

(A) 0
(B) 1
(C) n-1
(D) n

Answer 1

Question #22434 Let $U$ and $V$ be vector spaces over a field $F$ and $\dim U = n$. Let $T: U \to V$ be a linear operator, then $\mathrm{rank}(T) + \dim \ker(T) = \ldots$

(A)0

(B) 1

(C) $n - 1$

(D) $n$

Please explain

Solution. By definition! nullity of $T$ is $\dim \ker T$. Every book on linear algebra contains the following fact: if $T: U \to V$ is linear transformation between finite dimensional linear spaces, then $\mathrm{rank}(T) + \dim \ker(T) = \dim U$. Hence:

Answer D.