Definition

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A linear transformation is a function between vector spaces preserving the structure of the vector spaces.

Suppose $V$ and $W$ are vector spaces (of arbitrary dimension). A function $f:V\to W$ is (as we recall) a rule which associates to each ${\bf v}\in V$ a unique vector $f({\bf v})\in W$ . We will call such a function a linear transformation if it commutes with the linear structure in the domain and range:

$\begin{eqnarray*} f({\bf v} + {\bf w}) = f({\bf v}) + f({\bf w})\qquad\forall {\bf v,w}\in V\\ f(\alpha{\bf v}) = \alpha f({\bf v})\qquad\qquad\forall \alpha\in\mathbb R, {\bf v}\in V \end{eqnarray*}$

In other words, $f$ takes sums to sums and scalar products to scalar products. These two properties can be combined into one, using linear combinations. Precisely,

Let $V,W$ be vector spaces. A function $f:V\to W$ is a linear transformation if $\begin{equation} f(\alpha{\bf v} + \beta{\bf w}) = \alpha f({\bf v}) + \beta f({\bf w})\qquad\forall \alpha,\beta\in\mathbb R, {\bf v}, {\bf w}\in V \end{equation}$

Show that $f$ is a linear transformation iff $f(\alpha _1{\bf v}_1 +\dots \alpha _n{\bf v}_n) = \alpha _1 f({\bf v}_1) +\dots \alpha _n f({\bf v}_n)\qquad \forall \alpha _i\in \mathbb R, {\bf v}_i\in V$

As a consequence of this exercise, we have

If $f: V\to W$ is a linear transformation, then it is uniquely determined by its values on a basis for $V$ . Conversely, if $S := \{{\bf v}_1,\dots {\bf v}_n\}$ is a basis for $V$ and $g:S\to W$ is a function from the set $S$ to $W$ , then $g$ extends uniquely to a linear transformation $f:V\to W$ with $f({\bf v}_i) = g({\bf v}_i), 1\le i\le n$ . Thus, if $f_1, f_2:V\to W$ are two linear transformations which agree on a basis for $V$ , then $f_1 = f_2$ .

Linear transformations may be used to define subspaces. Let $L:V\to W$ be a linear transformation. The kernel of $L$ is then $ker(L) := \{{\bf v}\in V\ |\ L({\bf v}) = {\bf 0}\}\subset V$ The image of $L$ is defined as $im(L) := \{ {\bf w}\in W\ |\ \exists {\bf v}\in V \text { with } L({\bf v}) = {\bf w}\} \subset W$ The image of $L$ is sometimes denoted $L(V)$ . It is also referred to as the range of $L$ . These subspaces are useful in defining specific types of linear transformations. We say $L:V\to W$ is

surjective or an epimorphism iff $im(L) = W$ ;
injective or a monomorphism iff $ker(L) = \{\bf 0\}$ ;
bijective or an isomorphism iff $L$ is both surjective and injective.

We say that two vector spaces $V$ and $W$ are isomorphic if there exists a linear transformation $L:V\to W$ which is an isomorphism.

For any linear transformation $L:V\to W$ , $ker(L)$ is a subspace of $V$ and $im(L)$ is a subspace of $W$ . Moreover, $Dim(V) = Dim(ker(L)) + Dim(im(L))$

It is natural to ask: what does it mean for a map to be a linear transformation? For example, if $f:\mathbb R^n\to \mathbb R^m$ , or more generally if $f:V\to W$ satisfies the above property, does it admit some simple description? We investigate this question next.