# Linear algebra projection

## Projection matrix p^2=p proof

So we need to figure out some way to calculate this, or a more mathematically precise definition. So let's see if we can calculate a c. I think the shadow is part of the motivation for why it's even called a projection, right? We know that c minus cv dot v is the same thing. Let me draw my axes here. Now, one thing we can look at is this pink vector right there. The red line in the plot below is subset defined by points satisfying with. We also know that this pink vector is orthogonal to the line itself, which means it's orthogonal to every vector on the line, which also means that its dot product is going to be zero.

Let me do this particular case. Let me draw my axes here. So let's say that this is some vector right here that's on the line. Well, the key clue here is this notion that x minus the projection of x is orthogonal to l.

## Orthogonal projection vector

This is a scalar still. Now, let's say I have another vector x, and let's say that x is equal to 2, 3. We could write it as minus cv. So multiply it times the vector 2, 1, and what do you get? Even though we have all these vectors here, when you take their dot products, you just end up with a number, and you multiply that number times v. So all of that over 2, 1, dot 2, 1 times our original defining vector v. You just kind of scale v and you get your projection. Now, we also know that x minus our projection is orthogonal to l, so we also know that x minus our projection-- and I just said that I could rewrite my projection as some multiple of this vector right there. So let me draw that. For the explanation of projections, we will use the standard matrix algebra notation for points: is a point in -dimensional space and is smaller subspace. I haven't even drawn this too precisely, but you get the idea. The projection of x onto l is equal to what?

So times the vector, 2, 1. It almost looks like it's 2 times its vector. When you take these two dot of each other, you have 2 times 2 plus 3 times 1, so 4 plus 3, so you get 7. Here we provide a quick review and then provide some data analysis related examples. You're beaming light and you're seeing where that light hits on a line in this case.

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