Null Space and Range

Definition: A linear transformation $T : V \to W$ . The null space of $T$ , denoted as $N (T)$ or $null T$ it is the set of all the vectors $v$ of $V$ such that $T (v) = 0$ , that is $N (T) = {v \in V : T (v) = 0}$ . $N (T)$ is a subspace of $V$ .
Theorem: $T \in L (V, W)$ , $T$ is injective iff $N (T) = {0}$

Definition: A linear transformation $T : V \to W$ . The range (or image) of $T$ , denoted as $R (T)$ is a subspace of $W$ . $R (T) = {T (v) : v \in V}$ .

Theorem: Given a basis $β$ of $V$ , and $T : V \to W$ linear, then $R (T) = span (T (β))$ .

Definition: Given a linear transformation $T : V \to W$ , $dim (N (T))$ is called the nullity also denoted as $n (T)$ or $nullity (T)$ , and, $dim (R (T))$ is called the rank of $T$ , denoted as $r (T)$ or $rank (T)$

Rank-Nullity Theorem

Let $V$ and $W$ be vector spaces, and $T : V \to W$ be linear. If $V$ is finite-dimensional, then:

n (T) + r (T) = \dim (V)

Corollary: Let $V$ and $W$ be finite-dimensional vector spaces where $\dim (V) > \dim (W)$ , then any $T \in L (V, W)$ cannot be injective.

Corollary: Let $V$ and $W$ be finite-dimensional vector spaces where $\dim (V) < \dim (W)$ , then any $T \in L (V, W)$ cannot be surjective.

Corollary: Let $V$ and $W$ be finite-dimensional vector spaces where $\dim (V) = \dim (W)$ , and any $T \in L (V, W)$ , $T$ is injective iff $T$ is surjective.

Theorem: Let $V$ and $W$ be finite-dimensional vector space over the field $F$ , suppose the set ${v_{1}, \dots, v_{n}}$ be a basis for $V$ . For $w_{1}, w_{2} \dots w_{n} \in W$ , there exists exactly one $T \in L (V, W)$ , such that $T (v_{i}) = w_{i},$ for all $1 \leq i \leq n$

Corollary: If for a basis ${v_{1}, \dots, v_{n}}$ of $V$ , and $T, U \in L (V, W)$ , and $T (v_{i}) = U (v_{i})$ for all $1 \leq i \leq n$ , then $U = T$ .