Essence of Linear Algebra (3): Matrices as Linear Transformations

Wed, 15 Jan 2025 09:00:00 +0000

The Big Idea#

Open a traditional textbook and matrices show up as “rectangular arrays of numbers.” You learn rules for adding and multiplying them, but no one explains why the multiplication rule looks the way it does, or why $AB \neq BA$ in general.

Here is the secret the symbol-pushing version hides: a matrix is a function that transforms space. Every $m \times n$ matrix is a machine that eats an $$n$$ -dimensional vector and spits out an $$m$$ -dimensional one. Once you can see that, the strange rules stop being strange. They are simply the bookkeeping for what happens to the basis vectors.

Linear Transformations on Chen Kai Blog

Essence of Linear Algebra (3): Matrices as Linear Transformations

The Big Idea#