Orthogonal basis

Imagine you have a big box of different toys. You want to organize the toys so that all the toys are easy to find and easy to sort.

To do this, you pick a few toys that are very different from each other. For example, you might choose a teddy bear, a soccer ball, and a toy car. These toys are called the "basis" of your toy box.

Now, you can put any other toy in the box into one of these three groups: it's either a teddy bear toy, a soccer ball toy, or a toy car toy.

Importantly, these groups don't overlap - a toy can't be both a teddy bear toy and a soccer ball toy at the same time.

This is like an "orthogonal basis" in mathematics. When we have a big set of numbers or functions, we can pick a few special ones that are very different from each other. Using these special numbers/functions, we can organize all the other numbers/functions into groups.

These groups don't overlap, just like the toy groups. And just like the toys, the special numbers/functions are called the "basis" of our set.