Mathematical Immaturity

2.1 Linear transformations

$\quad$ One of the ultimate goals of analysis is a comprehensive study of functions whose domains and ranges are subsets of linear spaces. Such functions are called transformations, mappings, or operators. This chapter treats the simplest examples, called linear transformations.

$\quad$Let $V$ and $W$ be two sets. The symbol \begin{align*} T:V \to W \end{align*} will be used to indicate that $T$ is a function whose domain is $V$ and whose values are in $W.$ For each $x$ in $V,$ the element $T(x)$ in $W$ is called the image of $x$ under T, and we say that $T$ maps $x$ unto $T(x).$ If $A$ is any subset of of $V,$ the set of all images $T(x)$ for $x$ in $A$ is called the image of $A$ under $T$ and is denoted by $T(A).$ The image of the domain $V,$ $T(V),$ is the range of $T.$

$\quad$ Now we assume that $V$ and $W$ are linear spaces having the same set of scalars, and we define a linear transformation as follows.

$\quad$ Definition. $\quad$ If $V$ and $W$ are linear spaces , a function $T: V \to W$ is called a linear transformation of $V$ into $W$ if it has the following two properties:
$\quad$ (a) $\quad T(x + y) = T(x) + T(y) \quad$ for all $x$ and $y$ in $V,$
$\quad$ (b) $\quad T(cx) = cT(x) \quad$ for all $x$ in $V$ and all scalars $c.$

In other words, $T$ preserves addition and multiplication by scalars. The two properties can be combined into one formula that states for all $x, y$ in $V$ and all scalars $a$ and $b:$ $$ T(ax + by) = aT(x) + bT(y).$$ By induction, we have the more general relation: $$T\left(\sum_{i=1}^n a_ix_i\right) = \sum_{i=1}^n a_iT(x_i)$$ for any $n$ elements $x_1, \dots, x_n$ in $V$ and any $n$ scalars $a_1, \dots, a_n.$