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?
• why is it so important to bring in w? like... we bring it in only to kill it!