0. Characterizations

Definition. $T:$   R$^n \rightarrow$   R$^m$ is a linear map if

(i) $T($v$+$   w$) = T($v$) + T($w$)$, for any v$,$   w (``additivity''), and

(ii) $T(r$   v$) = rT($v$)$, for any vector v and scalar $r$ (``homogeneity'').

Example. Let $A$ be an $m \times n$ matrix, and let $T _ A$ be defined by $T _ A($x$) = A$   x (where x is any column vector). Then $T _ A$ is a linear map, by the algebraic properties of matrix operations.

Observations. If $T$ is a linear map then

(a) $T($0$) =$   0 (the origin goes to the origin),

(b) $T(r$   v$+ s$   w$) = rT($v$) + s T($w$)$ ($T$ is compatible with linear combinations).

Outline of proof. Use the standard basis of R$^n$: e$_ 1 = \left[\begin{array}{c}1\\ 0\\ {\dots }\\ 0\end{array}\right], \dots ,\$   e$_ n = \left[\begin{array}{c}0\\ {\dots }\\ 0\\ 1\end{array}\right]$. For any $T$, let $A$ be the matrix whose $i$-th row is $T($e$^{(i)})$; then $T($x$)$ is the same as $A$   x for x$=$ one of the e$^{(i)}$ and so for any x, since every vector x in R$^m$ is a linear combination of standard basis vectors and $T$ preserves linear combinations. Thus $T = T _ A$.

Remarks

(1) Linear maps leave the origin fixed. We'll also be discussing ``affine'' maps, which are a generalization in which the origin can move. Because these still take lines to lines, some texts also call affine maps ``linear''.

(2) If we're thinking about linear maps abstractly we'll use just the letter $T$; if we have $A$ in mind we'll use $T _ A$. By the Theorem these two points of view are equivalent. Later on we'll use $T$ for other kinds of maps as well.

(3) The definition of $T _ A($x$)$ as $A$   x assumes that the $n$-tuple x is represented as a row vector; this is a common assumption in computer graphics packages. If we use column vectors, it would be $A$   x. Notes in this course generally assume row vectors.

(4) In projective geometry, it is shown that any one-to-one maps of R$^n$ onto itself taking lines to lines and the origin to the origin must be a linear map. (This is a deep fact.)