Orthogonal group

Hello there! Today, we are going to talk about something called the orthogonal group.

Imagine you have some shapes, like squares or triangles. We often move these shapes around to make patterns or to fit them into certain places. But sometimes, if we rotate or flip these shapes, they end up looking different than before.

The orthogonal group is a group of matrices that help us keep these shapes looking the same when we move them around. Instead of rotating and flipping them, we use these matrices to move the shapes without changing their appearance.

Now, what are matrices? Think of them as boxes with numbers inside. Depending on the numbers inside the box, we can do all sorts of calculations with them.

The matrices in the orthogonal group are special because they have certain properties that make them very useful for moving shapes without changing them. For example, these matrices have rows and columns that are all perpendicular (or "orthogonal") to each other.

So, if we have a square that we want to move around without changing its shape, we can use one of these matrices from the orthogonal group. By multiplying the square's vertices (or "corner points") with a matrix from the orthogonal group, we can move the square without changing its angles or sides.

That's the basic idea of the orthogonal group. It's all about using special matrices to move shapes around without distorting them. Hope that helps!