Linear Transformations and Their Matrices

A matrix $A$ has served as a bookkeeping device for a system. Read the other way, $A$ acts on a vector $x$ by multiplication, sending it to a new vector $A x$ ; solving $A x = b$ becomes finding which inputs $A$ sends to $b$ . This viewpoint gathers the earlier results (existence, uniqueness, span, independence) around a single map.

Transformations

A transformation (equivalently function or mapping) $T$ from $R^{n}$ to $R^{m}$ assigns to each $x \in R^{n}$ a vector $T (x) \in R^{m}$ , written $T : R^{n} \to R^{m}$ . This is the ordinary idea of a function, lifted from numbers to vectors.

The range can be smaller than the codomain: the codomain is the space in which images are permitted, the range the set of images actually attained.

Matrix transformations

The first source of transformations is matrix multiplication. Fix an $m \times n$ matrix $A$ and define $T (x) = A x$ , often written $x \mapsto A x$ . The domain is $R^{n}$ (matching the columns of $A$ ), the codomain is $R^{m}$ (matching the rows), and the range is the span of the columns of $A$ , because every image $A x$ is a linear combination of those columns.

A transformation T sends each x in the domain to its image T(x) in the codomain; the range is the set of all images actually reached.

For example, $A = 100010000$ sends $(x_{1}, x_{2}, x_{3})$ to $(x_{1}, x_{2}, 0)$ : a projection of $R^{3}$ onto the $x_{1} x_{2}$ -plane. The map $A = [1031]$ is a shear that slides the top of a square rightward while fixing the base, turning it into a parallelogram.

Linear transformations

Two properties, already proved for $x \mapsto A x$ , single out the central class of maps.

Linearity means the map commutes with the two vector operations: adding then transforming equals transforming then adding, and likewise for scaling. Every matrix transformation is linear, by the linearity of $x \mapsto A x$ . Two consequences follow immediately for any linear $T$ :

T (0) = 0, T (c u + d v) = c T (u) + d T (v) .

The first comes from $T (0) = T (0 \cdot u) = 0 \cdot T (u) = 0$ . The second, applied repeatedly, gives the general rule

T (c_{1} v_{1} + \dots + c_{p} v_{p}) = c_{1} T (v_{1}) + \dots + c_{p} T (v_{p}),

known in physics and engineering as the superposition principle: the response to a combination of inputs is the same combination of the responses. A map satisfying the two-vector rule for all $u, v, c, d$ is linear (set $c = d = 1$ , then $d = 0$ ), so that rule can serve as the test.¹

A contraction or dilation $T (x) = r x$ is linear for any scalar $r$ . By contrast, $T (x_{1}, x_{2}) = (4 x_{1} - 2 x_{2}, 3∣ x_{2} ∣)$ is not: the absolute value breaks scaling, since $T (- u) \neq = - T (u)$ in general. An affine map $x \mapsto A x + b$ with $b \neq = 0$ also fails, because $T (0) = b \neq = 0$ .

The standard matrix

A linear transformation is determined by where it sends the coordinate axes. Let $e_{1}, \dots, e_{n}$ be the columns of the $n \times n$ identity — the standard basis vectors. Any $x = (x_{1}, \dots, x_{n})$ is $x = x_{1} e_{1} + \dots + x_{n} e_{n}$ , so by linearity

T (x) = x_{1} T (e_{1}) + \dots + x_{n} T (e_{n}) .

The images of the basis vectors determine every image. Packing them as columns recovers a matrix.

So linear transformation and matrix transformation describe the same objects; the first names a property, the second names the implementation.² To build $A$ , transform the axes and read off columns.

A linear map is fixed by the images of e_1 and e_2; those images become the columns of the standard matrix A.

The geometric maps of the plane

The common $2 \times 2$ maps are catalogued by their action on the unit square, all determined by the images of $e_{1}$ and $e_{2}$ .

Transformation	Standard matrix	Effect on the unit square
Reflection through the $x_{1}$ -axis	$[10 0 - 1]$	flips vertically
Reflection through $x_{2} = x_{1}$	$[0110]$	swaps coordinates
Horizontal contraction / expansion	$[k 0 01]$	scales width by $k$
Horizontal shear	$[10 k 1]$	slants into a parallelogram
Projection onto the $x_{1}$ -axis	$[1000]$	collapses onto a line
Rotation by angle $φ$	$[cos φ sin φ - sin φ cos φ]$	turns about the origin

A rotation about the origin carries the unit square (dashed) to its image; e_1 and e_2 land on the columns of the rotation matrix.

Three standard maps acting on the unit square (dashed): a horizontal shear, a reflection through the x_1-axis, and a projection onto the x_1-axis.

Onto and one-to-one

The existence and uniqueness questions of the whole chapter become two properties of the map $T$ .

Onto asks whether every target is hit (existence); one-to-one asks whether no target is hit twice (uniqueness).

Both reduce to pivot conditions on the standard matrix.

The onto criterion is the spanning theorem reworded. For one-to-one, if $T (u) = T (v)$ then $T (u - v) = 0$ ; distinct inputs with the same image give a nonzero solution of $T (x) = 0$ , and the independence of the columns says no such solution exists.³

The pivot counts fix the possibilities by shape. A map $R^{n} \to R^{m}$ can be:

Shape	Onto possible?	One-to-one possible?
$n < m$ (taller)	no — too few columns to span	yes
$n = m$ (square)	yes	yes — and one holds iff the other does
$n > m$ (wider)	yes	no — a free variable is forced

A projection ( $[1000]$ ) is neither onto nor one-to-one, and a rotation is both.

Lay, Linear Algebra and Its Applications, §1.8 — Introduction to Linear Transformations: the domain/codomain/range vocabulary, matrix transformations, the definition of linearity, and the consequences $T (0) = 0$ and the superposition principle. ↩
Lay, §1.9 — The Matrix of a Linear Transformation: every linear $T : R^{n} \to R^{m}$ has a unique standard matrix whose columns are $T (e_{j})$ , and the catalogue of geometric maps of the plane. ↩
Lay, §1.9, Theorems 11 and 12: $T$ is one-to-one iff $T (x) = 0$ has only the trivial solution, iff the columns of $A$ are independent; $T$ is onto iff the columns of $A$ span the codomain. ↩