Let $V$ and $W$ be vector spaces over a field $\mathbb{F}.$ We call a function

$$T:V\to W$$ a linear transformation from $V$ to $W$ if for all $\mathbf{x},\mathbf{y}\in V$ and all $c\in \mathbb{F},$ we have

1. (a) $T(x+y)=T(x)+T(y)$ and
2. (b) $T(cx)=cT(x)$

The first condition is called additivity and the second is called homogeniety.

Let $V$ and $W$ be vector spaces and let $T:V\to W$ be linear.

We define the null space (also: kernel) $N(T)$ of $T$ to be the set of all vectors $x\in V$ such that $T(x)=0,$ that is, $N(T)=\{\mathbf{x}\in V:T(\mathbf{x})=0.\}$

We define the range of a linear transformation (also: image of a linear transformation) $R(T)$ of $T$ to be the subset of $W$ consisting of all @images (under T) of vectors in $V,$ that is, $R(T)=\{T(\mathbf{x}):\mathbf{x}\in V\}.$

Let $V$ and $W$ be vector spaces and let $T:V\to W$ be linear. If $N(T)$ and $R(T)$ (null space and range of $T$, respectively) are @finite dimensional, the we define the nullity of $T,$ denoted $\text{nullity}(T),$ and the rank of $T,$ denoted $\text{rank}(T),$ to be the @dimensions of $N(T)$ and $R(T),$ respectively.