a projection is amap onto a subspace, that sends every vector to a vector in the subspace, and leaves vectorsd that are already in the subspace where they are.
so if V is a vector space and X is a subspace we want to project on, and if
Y is any complementary subspace, i.e. X and Y together generate V, and X and Y have only the zewro vector in common, then every vector in V can be written uniquely in the form x+y where x is in X and y is in Y.
Then the map f sending x+y to x, is a projection onto X, "along" Y.
Notice that f(f(v)) = v for all v, since once v gets into X it stays put. And also Y = ker(f), since vectors in Y go to zero.
Indeed any linear map f such that f^2 = f is asuch a projection.
f is called an "orthogonal" projection if Y = ker(f) is orthogonal to X = im(f).\
at least i think so, you should of course verify everything by proving all these either trivial or false statements.