Matrices as transformations

Intuitive problem

Solving systems of linear equations leaves a residue. The unknowns get their values and are forgotten, but the grid of coefficients keeps turning up with a life of its own: two problems with different totals but the same grid yield to the same eliminations in the same order. An array that behaves this consistently is asking to be treated as a single object. What is that object? The grid is a machine: feed it any point of the plane and it returns another point. The machine image breaks in one place, and the break is the whole point: a physical machine can respond to inputs however it likes, while this one is bound by a constraint so tight that two sample outputs determine its behavior everywhere. This entry reads the array as the record of an action on space, and the multiplication rule usually handed down for memorization falls out of that reading.

Definition

Work in the plane; higher dimensions repeat the same story with more columns. A vector v is an arrow from the origin, named by its coordinates: v = (x, y) means x steps east and y steps north, in the language of the entry on vectors as directed quantities. A linear transformation T is a map sending each vector v to an output T(v) under two constraints: the origin stays put, and combinations are preserved, meaning T(a v + b w) = a T(v) + b T(w) for any vectors v and w and any numbers a and b. Geometrically, such a map keeps straight lines straight and keeps evenly spaced grid lines evenly spaced. Rotations about the origin, stretches, and shears qualify; a translation moves the origin, so it does not.

Write e1 = (1, 0) and e2 = (0, 1) for the two basis arrows, one step east and one step north. A matrix is the record of a linear transformation: an array whose first column is T(e1) and whose second column is T(e2), the landing places of the two basis arrows. That is the whole definition. To read an unfamiliar matrix, watch where its columns say the basis arrows land.

Derivation

First, recover the action on an arbitrary vector from the two recorded landings. Any v = (x, y) is a combination of basis arrows: v = x e1 + y e2. Preservation of combinations then forces T(v) = x T(e1) + y T(e2). In words, the output is x copies of the first column plus y copies of the second. Define matrix-vector multiplication to be exactly this weighted sum of columns, written A v where A is the matrix recording T; the row-by-column arithmetic taught as a rule is this sum ground out coordinate by coordinate. Two recorded arrows determine the transformation everywhere, which is the promised tightness of the machine's constraint.

Next, composition. Take two linear transformations recorded by matrices A and B, and perform B first, then A. The composite is itself linear, since each stage fixes the origin and preserves combinations, so it has its own record. Its columns are the landing places of the basis arrows under the two-stage action: B sends e1 to B's first column, and then A acts on that column the way it acts on any vector. So the composite's first column is A applied to B's first column, and likewise for the second. Define the matrix product AB to be this record of the composition, B's action followed by A's. The familiar multiplication rule is this definition computed in coordinates.

Two structural facts now cost nothing. The product is associative, (AB)C = A(BC) for any third matrix C, because both sides record the same three-stage itinerary, C then B then A: one composite map, so one matrix. And the product has no reason to commute, because doing B then A and doing A then B are different itineraries through space; the worked example below exhibits a pair whose two orders land the same arrow in visibly different places.

The do-nothing action, sending every vector to itself, is recorded by the identity matrix I, its columns e1 and e2 unchanged, with AI = IA = A. An inverse of A, written A^-1, is the record of the undo action when one exists: the transformation returning every landed arrow to where it started, so that A^-1 A = I.

One further quantity earns its sentence here because a mistake below depends on it. The determinant of A, written det A, is the factor by which the action scales area: the unit square spanned by e1 and e2 lands on the parallelogram spanned by the columns, and the size of det A is that parallelogram's area, with a negative sign when the action flips the plane over. A determinant of zero means the parallelogram has no area at all: the action has collapsed the plane onto a line or a point, and a collapse cannot be undone, so no inverse exists.

Worked example

Two actions, chosen so that their two orders disagree. Matrices are written here as lists of rows.

The rotation R turns the plane a quarter turn counterclockwise. The east arrow e1 lands pointing north, at (0, 1); the north arrow e2 lands pointing west, at (-1, 0). Recording the landings as columns gives R = [[0, -1], [1, 0]].

The stretch S doubles east-west distances and leaves north-south distances alone: e1 lands at (2, 0) and e2 stays at (0, 1), so S = [[2, 0], [0, 1]]. The stretch is deliberately lopsided, acting along one axis only; the limits section says why that choice matters.

Apply each to the sample vector v = (1, 1). The stretch gives S v = one copy of (2, 0) plus one copy of (0, 1), which is (2, 1); the rotation gives R v = (0, 1) + (-1, 0) = (-1, 1).

Now compose in both orders and track e1. Stretch then rotate, the product RS: S sends e1 to (2, 0), and R turns that arrow to (0, 2), a long arrow pointing north. Rotate then stretch, the product SR: R sends e1 to (0, 1), a vertical arrow, which the east-west stretch leaves untouched at (0, 1), a short arrow pointing north. Grinding out both products confirms the picture: RS = [[0, -1], [2, 0]] while SR = [[0, -2], [1, 0]]. The two composites disagree about where e1 lands, (0, 2) against (0, 1), so the two orders are different transformations.

Limits and boundary conditions

Only linear maps get a matrix in the sense of this entry. A translation moves the origin, and a map that bends lines breaks combinations; neither has a record here. The treatment is finite-dimensional: the plane, and its higher-dimensional cousins by the same column-recording move, with nothing said about infinite-dimensional spaces.

A singular matrix, one with determinant zero, collapses the plane onto a line or a point. The collapse merges distinct inputs onto shared outputs, so the information needed to undo it is gone, and no inverse matrix exists.

One boundary fact is instructive enough to state on its own. A uniform scaling, the same factor applied on every axis, commutes with every rotation: enlarging all directions alike is indifferent to how the plane has been turned, so for that pair the order genuinely does not matter. This is exactly why the worked example stretches one axis only; a demo built on uniform scaling would show the two orders agreeing and prove nothing. Failure to commute is the general situation, and commuting pairs like this one are special cases with a structural reason behind them.

Common mistakes

First, treating a matrix as a spreadsheet, an inert table of numbers to be stored and shuffled by rote. The table format is real, but the reading that makes the algebra sensible is columns as landing places; without it the multiplication rule looks arbitrary and non-commutation looks like a bug. Second, assuming the product commutes, by analogy with ordinary numbers. Pairs with AB = BA exist, uniform scalings among them, but they are the exception to be checked rather than the rule to be assumed. Third, treating a zero determinant as a minor computational detail, one number among many that happens to come out zero. It is the signal that the action has collapsed the plane, that distinct inputs have been merged, and that no matrix can undo the damage.

Build with it

Two tasks, one for each direction of the reading.

First, read an action. The matrix [[0, 1], [1, 0]], written as rows, arrives with no label. Track the two basis arrows through it, using columns as landing places, and name in plain words the geometric action it performs.

Second, write an action, then compose. Construct the matrix F for reflection across the x axis, the flip sending every point to its mirror image across that axis. Then take the quarter-turn counterclockwise rotation R from the worked example and compute both products: RF, which is reflect then rotate, and FR, which is rotate then reflect. Report both resulting matrices and identify the geometric action each composite performs, again by tracking the basis arrows.

Success means three things. The unlabeled action in the first task is named correctly. The constructed F is correct, checked by confirming its columns are the landing places the flip demands. And both compositions in the second task are computed, with the difference between the two orders stated explicitly; the two products disagree, and saying how is the evidence that the itinerary matters.