The Core Idea of Linear Algebra

“Algebra” roughly translates to “relationships” and linear means “line-like,” thus “Linear Algebra” means “line-like relationships”.

Algebra

So why does algebra from school seem to be just about letters rather than numbers.

In my head algebra is just about arranging and rearranging letters according to specific rules. However here is a way of looking at it.

Consider the equation below. You don’t know what value of the x and the y is.

(x + y)² = x² + 2xy + y²

However we can still workout the relationship between (x + y)² and x² + 2xy + y². We know that both expressions are equal, and we know the relation between the value of x and y and each side of the equation. We could write this as a function:

f(x, y) = (x + y)²

Linear

Linear relations are relation similar to what you can describe on a line. Linear relations are predictable. You double the input and you get double the output.

This is not a linear relation

f(x) = x²

If I double the input, the output is quadrupled. This is an example of a linear relation:

f(x) = 4x

However this is not a linear relation, even if you make think so.

f(x) = 2x + 4