Null Space and Range

Definition: A linear transformation $T:V\to W$. The null space of $T$, denoted as $N(T)$ or $\text{null }T$ 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: $T\in \mathcal{L}(V,W)$, $T$ 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 $\beta$ of $V$, and $T:V\to W$ linear, then $R(T)=\text{span}(T(\beta ))$.

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

Rank-Nullity Theorem

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

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

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

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

Theorem: Let $V$ and $W$ be finite-dimensional vector space over the field $\mathbb{F}$, 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 $T\in \mathcal{L}(V,W)$, such that $T(v_i)=w_i,$ for all $1\le i\le n$

Corollary: If for a basis $\{v_1,\dots ,v_n\}$ of $V$, and $T,U\in \mathcal{L}(V,W)$, and $T(v_i)=U(v_i)$ for all $1\le i\le n$, then $U=T$.