# Matrices and linear maps

A linear map makes each output a sum of the inputs, each multiplied by a fixed number, and a matrix is the table of those multipliers. The change from one frame to another at 0.6c is such a map. Writing it as a matrix is a later aid: the 1905 paper writes it as two equations.

Written for this edition, not translated from Einstein. Editorial review pending.

## How can a table of four numbers turn one frame's coordinates into another's?

Some rules turn inputs into outputs by multiplying and adding, and nothing else: $x' = p\,x + q\,t$ and $t' = r\,x + s\,t$, with the same four numbers p, q, r and s for every input. Such a rule is a linear map. Set the four numbers in a table, one row per output, and the table is its matrix. Each row says how much of each input goes into that output.

$$
\begin{aligned} x' &= p\,x + q\,t \\ t' &= r\,x + s\,t \end{aligned}
$$

x prime equals p x plus q t, and t prime equals r x plus s t.

The change from a frame at rest to one moving at v along x is a linear map: $x' = \gamma(x - vt)$ and $t' = \gamma(t - vx/c^2)$, with $\gamma = 1/\sqrt{1 - v^2/c^2}$. The relativity paper prints this factor as β; here it is written γ, the modern letter, and the ratio v/c is written out. At v = 0.6c, γ = 1.25 and γv = 0.75c, so the table of multipliers is:

$$
\left(\begin{aligned} 1.25 &\quad -0.75c \\ -0.75/c &\quad 1.25 \end{aligned}\right)
$$

A two by two table: first row 1.25 and minus 0.75 c; second row minus 0.75 over c and 1.25.

Two numbers read off the table describe what the map does. The determinant, ps − qr, is the factor by which it scales areas in the plane of x and t; for this map it is $1.25^2 - 0.75^2 = 1$, so areas are kept. The eigenvectors are the directions the map only stretches. For this map they are the two light lines, $x = ct$ and $x = -ct$, stretched by 0.5 and by 2: a flash sent either way still moves at c in the new frame.

The older Galilean rule, $x' = x - vt$ and $t' = t$, is also linear, with the table $p = 1$, $q = -v$, $r = 0$, $s = 1$, and its determinant is also 1. What separates the two is the light lines. The Galilean map sends $x = ct$ to $x' = (c - v)t$: the flash now moves at c − v, not c.

Writing the transformation as a matrix, with determinants and eigenvectors, is a later aid, tied to the geometry Minkowski gave the theory in 1908. §3 of the paper writes the transformation as equations and derives them from light signals and clocks; the table adds a way of reading them, not a premise.

## Worked example: Two simultaneous events, seen from 0.6c

1. Two events happen at the same time, 10 light-seconds apart along x: $\Delta x = 10$ light-seconds and $\Delta t = 0$.
2. The first row gives $\Delta x' = 1.25 \times 10 + (-0.75c) \times 0 = 12.5$ light-seconds.
3. The second row gives $\Delta t' = (-0.75/c) \times 10$ light-seconds $+\ 1.25 \times 0$. Ten light-seconds divided by c is 10 seconds, so $\Delta t' = -7.5$ s.
4. In the moving frame the events are not simultaneous: the one farther along x happened 7.5 seconds earlier.

## Where this lesson stops

A matrix is a table of multipliers applied to inputs. This lesson stops at two inputs and two outputs.

## This lesson builds on

- [Vectors and their components](/foundations/vectors-components/)
- [Functions and graphs](/foundations/functions-graphs/)
- [Quantities and units](/foundations/quantities-units/)