User:IssaRice/Linear algebra/How to remember the projection formula

yooo i keep forgetting this but i think i finally figured out a good mnemonic

let's recall the setting. we have some vector u, and we're trying to project it onto another vector v. it doesn't make sense to project onto the zero vector, so we require that v != 0.

the projection will live in the subspace spanned by v, i.e. the projection will have the form tv for some constant t.

The BIG idea #1 is to bring in a third vector w, orthogonal to v, so that u = tv + w.

The BIG idea #2 is to take the dot product of this equation using v, so we have u.v = t(v.v); now t = (u.v)/(v.v).

questions:

• above we took (.v) instead of (v.); but these are different in complex vector spaces. does that mean there are two orthogonal projections in complex vector spaces? oh actually it looks like you get the same result regardless of how you do it: https://math.stackexchange.com/a/1041974/35525
• why is it so important to bring in w? like... we bring it in only to kill it! (we could do it a bit artificially, by saying that we want to project u onto v, so write the projection as tv for some constant t. then by virtue of it being a projection, we must have u-tv orthogonal to v, so (u-tv).v=0. that will give us t as well. but we're still implicitly "talking about" w.)