Orthonormal basis

Imagine that you have a big collection of toys, and you need to find a way to put them away properly so that you can easily find them again later. You decide to organize them by size and color, but you need to choose a system to do this.

An orthonormal basis is like a very specific system for organizing your toys. It's a way to choose a group of special toys that are all different from each other, but when you put them together in the right order, they can represent any other toy in your collection.

Let's say that you have ten different toys, and you want to pick four toys to be your orthonormal basis. You choose a red ball, a blue square, a green triangle, and a yellow star. Each of these toys is unique and different from the others, but they also have some very specific things in common.

First, they're all "orthogonal," which means that they're all at right angles to each other. This helps you make sure that none of your toys overlap or get mixed up with each other.

Second, they're all "normalized," which means that they're all the same size and shape. This helps you make sure that your toys are all consistent and easy to understand.

Now, when you want to organize the rest of your toys, you can use combinations of these four special toys to represent any other toy in your collection. For example, you could describe a purple circle as "half red ball plus half blue square," or a pink star as "one-third red ball plus two-thirds yellow star."

By using this orthonormal basis, you've created a special system for organizing your toys that makes them easy to understand and work with.