Linear Algebra is the math of superposition

Friday September 18, 2026

If math is about patterns, what patterns is Linear Algebra about? Using the word “linear” again feels circular, and still leaves us needing a special definition. Let's borrow “superposition,” as it's used in physics and engineering, to name the patterns of Linear Algebra.

A beautiful pattern

Emmy started growing flowers at home, using special lights and soil. She keeps all the flowers she grows, pressing them inside a large book to preserve them. When she remembers, at the end of the month she records the number of flowers she's grown since she started.

| at end of month | total flowers grown |
|-----------------|---------------------|
| 3               | 6                   |
| 5               | 10                  |
| 7               | 14                  |

If things keep going this way, how many flowers will Emmy have after twelve months? How many ways can you find the answer?

Method one: Multiply month three by four

Twelve is four times three. At the end of month three, there were six flowers. Multiply six by four and get 24 flowers at the end of month twelve.

Method two: Add months five and seven

Twelve is five plus seven. At the end of month five there were 10 flowers, and at the end of month seven there were 14. Add 10 and 14 and get 24 flowers at the end of month twelve.

Method three: Multiply twelve by two

The number of flowers is always double the month number. Multiply twelve by two and get 24 flowers at the end of month twelve.

That multiplier, two, goes by many names. In this case it is a singular value and arguably also an eigenvalue—though these are not always the same, and there's more to say.

Superposition

When these methods work, the pattern satisfies superposition. Superposition refers to putting multiple things “in the same place,” adding them up. In a somewhat special sense of linear, a system with superposition is also called linear. It might be best to think of linear as short for linear superposition. Linear Algebra is the mathematics of superposition, the mathematics of linear systems in this sense.

Superposition is usually defined based on method one (multiplying by a number works) and method two (adding things together works). Method three follows from those two. Much of Linear Algebra is about extending method three to more complicated cases.

You may have noticed that many patterns don't appear to follow superposition. Why does Linear Algebra focus on such simple patterns?

One reason is that by keeping the patterns simple in this way, it's possible to make them more complex in another way: by adding dimensions. Linear Algebra extends to many dimensions, becoming the math of data.

Another reason is that although the rules of linear superposition seem like a strong constraint on patterns, lots of complex patterns can still be represented or at least approximated.