🔀Linear Transformations

Lens-keeper Ash does not think about one vector at a time. She thinks about what a map does to everything at once — and the only maps worth a lens are the ones that leave the ruled grid of the isle looking like a ruled grid afterwards: lines stay lines, the origin stays put, and evenly spaced marks stay evenly spaced.

Written down, that comes to two rules. A map T is a linear transformation when, for every pair of vectors u and v and every number c:

• ◆it keeps sums: T(u + v) = T(u) + T(v)
• ◆it keeps scalar multiples: T(cu) = cT(u)

Between them, those two say that whatever you do to the inputs — add them, stretch them, mix them — you may just as well do it to the outputs instead. That is the whole idea, and everything else in this craft is a consequence of it.

Here is the cheapest consequence, and the first test to run every time. Put c = 0 in the second rule: T(0u) = 0T(u), which reads T(0) = 0. A linear map must send the zero vector to the zero vector. So if a formula carries a constant on the end — T(x, y) = (x + 4, y) — it moves the origin, and it is not linear. One line of arithmetic settles it.

The second test is nearly as cheap. Take u = (1, 1) and compare T(2u) with 2T(u). A squared term or a product of two variables will not survive it: if the first component is x² then doubling the input multiplies that component by 4, not by 2.

Between them the two tests catch everything you will meet. A map is linear exactly when every component is a sum of multiples of the variables — no constants, no powers, no products of two variables, nothing under a square root.

This is the theorem the whole subject turns on. Take any vector x = (x1, x2) and split it up using the standard basis vectors e1 = (1, 0) and e2 = (0, 1):

x = x1(1, 0) + x2(0, 1) = x1*e1 + x2*e2

Now push it through a linear map and let the two rules do the work:

T(x) = T(x1*e1 + x2*e2) = x1*T(e1) + x2*T(e2)

Read that carefully. It says T is completely decided by two vectors: what it does to e1 and what it does to e2. Know those, and you know what T does to every vector in the plane, for ever.

So we store them. Stand T(e1) up as the first column and T(e2) up as the second, and you have the standard matrix A of T, with T(x) = Ax:

A = [[first entry of T(e1), first entry of T(e2)], [second entry of T(e1), second entry of T(e2)]]

Images stand up as columns, never lying down as rows. Writing them as rows gives the transpose, which is a different map, and it is the single most common slip in this section.

Working out Ax. Each entry of the answer comes from one whole row of A meeting the whole of x: pair the entries up in order, multiply, and add.

[[2, -1], [3, 4]] applied to (5, 1) = ((2)(5) + (-1)(1), (3)(5) + (4)(1)) = (9, 19)

The shapes have to agree: A must have as many columns as x has entries, and the answer has as many entries as A has rows. A 2 by 3 matrix eats a vector of R3 and hands back a vector of R2.

A handful of maps of the plane come up so often that their matrices are worth recognising on sight. Every one of them is built the same way — send (1, 0) somewhere, send (0, 1) somewhere, stand the two images up as columns:

• ◆reflection in the x-axis: (1, 0) stays, (0, 1) flips to (0, -1), so A = [[1, 0], [0, -1]]
• ◆reflection in the y-axis: A = [[-1, 0], [0, 1]]
• ◆reflection in the line y = x: the coordinates trade places, so A = [[0, 1], [1, 0]]
• ◆rotation of 90 degrees anticlockwise: (1, 0) swings up to (0, 1) and (0, 1) swings round to (-1, 0), so A = [[0, -1], [1, 0]]
• ◆rotation of 180 degrees: A = [[-1, 0], [0, -1]]
• ◆rotation of 270 degrees anticlockwise: A = [[0, 1], [-1, 0]]
• ◆scaling by k about the origin: A = [[k, 0], [0, k]]
• ◆horizontal shear sending (0, 1) to (k, 1): A = [[1, k], [0, 1]]

The two rotations are the pair worth being careful about: they are transposes of each other, so writing the images as rows turns a quarter-turn one way into a quarter-turn the other way. If you are unsure, draw the arrow (1, 0), turn it, and look at where its tip lands.

Notice that none of these has a constant anywhere. A translation — sliding the whole plane two steps to the right — is not a linear transformation at all, because it moves the origin. It is a perfectly good map, it is just not one a single matrix of this kind can hold.

Do R first and then S, and you have a new transformation T with T(x) = S(R(x)). Because R(x) is just Rx, that reads T(x) = S(Rx) = (SR)x: the matrix of a composition is the product of the matrices.

The order is the part to be careful with. The map that acts first stands on the right, next to x, because that is the one x meets first. And matrix multiplication does not commute, so SR and RS are usually different maps: rotate then reflect and you land somewhere else than reflect then rotate. Try it with your hands on a book cover.

Two more questions get asked of every transformation, and one row reduction answers both.

One-to-one asks whether two different inputs can ever share an image. They cannot exactly when every column of A carries a pivot — no column is left free, so nothing but the zero vector is sent to zero.

Onto asks whether every vector of the codomain can be reached. It can exactly when every row of A carries a pivot.

So count the pivots once, then compare that count with the number of columns and with the number of rows. Shapes settle some cases before you start: a map from R3 to R2 has more columns than rows and can never be one-to-one, and a map from R2 to R3 can never be onto.