projection কী?

Example 20.16 (A projection and its symmetry, explicitly)

In $\R^2$, project onto $F = \operatorname{Vect}(1,1)$ along $G = \operatorname{Vect}(0,1)$. Decompose $(x, y) = a(1,1) + b(0,1)$: the first coordinate gives $a = x$, the second $b = y - x$. Hence

Check the algebra: $p(p(x,y)) = p(x,x) = (x,x)$, and $s(s(x,y)) = s(x, 2x - y) = (x, 2x - (2x - y)) = (x, y)$. Geometrically, $s$ is the “oblique reflection” across the line $y = x$ in the vertical direction: it fixes $F$ pointwise and reverses $G$. Had we projected onto the same $F$ along $G' = \operatorname{Vect}(1,-1)$ instead, the formula would change to $p'(x,y) = \bigl(\frac{x+y}2, \frac{x+y}2\bigr)$: a projection is determined by its image and its kernel, never by the image alone.