What is the idea behind a projection operator? What does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me?

I don't need examples or the definition - I want to know why and how its need arose, and what is the idea behind it?

Solution 1:

This is how I used to imagine projections:

If a mouse:

gets run over by a steamroller:

It will look like this:

Now if it gets run over by a steamroller another time, it will still look like this:

Solution 2:

I do think the wiki's explanation is pretty good:

A projection is a mapping of a set (or other mathematical structure) into a subset 
(or sub-structure), which is equal to its square for mapping composition (or, 
in other words, which is idempotent).

Even more simply: it is an idempotent homomorphism.

Solution 3:

If you want a practical application of it I have one, it is indeed as they say that you take a higher dimensional object and put it into a lower dimensional one. A real world example is computer games. They are almost all generated in a 3D enviorement where they interact with polygons and all that computer stuff. However our screens are only 2D so all that 3D delicious game stuff must be projected onto a 2D surface to be displayed.

Solution 4:

I always think of it as mapping an object to its shadow (like a 3D person "projected" onto the plane of the ground a 2D image). You can get different projections by adjusting the location of the light source. Of course this is just 3D-2D "intuition" but it has worked for me...

Solution 5:

It projects the input onto a lower dimensional subset.

One example of a projection is the function

\begin{align} P:\mathbb{R}^3&\to\mathbb{R}^3\\ (x,y,z)&\mapsto (x,y,0) \end{align}

That is $P(x,y,z) = (x,y,0)$. You can visualize this function as it takes an input (a point in $3$D-space) and maps the point to the point below it on the $(x,y)$-plane (which is $2$-dimensional).