What is the Significance of (I-P)(x) in Linear Algebra?

[transgalactic]

i have here this solution to a problem in projection
i wrote it down
and added 2 questions in it

http://img134.imageshack.us/my.php?image=img8125du8.jpg

 

[HallsofIvy]

Your two questions are "Why are P(x_R)= x_R and P(x_N)= 0?" and "what is the meaning of (I- P)(x)?".

You are projecting x onto R. That is, x_R= P(x), the component of x lying along R. If some vector, y, is already pointing in the R direction, P(y)= y. If you project again, since x_R is already along R, projecting again just gives you the same thing again.

As for P(x_N)= 0, that is only true if N is orthogonal to R. Your picture does not make it look like that but you say "ker P= N". If x_N is in ker P, then by definition of kernel, P(x)= 0. Are you sure you aren't told somewhere that R and N are orthogonal?

Finally, (I- P)x= Ix- P(x)= x-P(x). It is x, with P(x) subtracted. If you drop a perpendicular from the tip of x to R, the intersection is at the tip of R- you have a right triangle with hypotenuse x, "near side" P(x), and opposite side x- P(x). If you "follow" the path from the origin of x to its tip, then down that perpendicular (call it "y"), you get to the tip of P(x): x+ y= P(x) so y= P(x)- x= -(I- P)x: -(I- P)x and so (I-P)x is always perpendicular to R.

 

[transgalactic]

can you please make a drawing to explain this p(Xr) p(Xn)
because i can't emagine all this components
and you said aslo that's my drawing is not correct

regarding the second part i didnt understand
the theoretical part of it
why are they doing
(I- P)(x)
what is the theoretical meenig of that

 

[HallsofIvy]

If you did not understand what I said before, then I recommend you talk to your teacher. Apparently you do not understand what a "projection" is.

 

[transgalactic]

correct me if i got it wrong p is taking the parts of the given vector
which is located on R

then the the axes are supposed to be perpandicular

i still can't understand the meening of the sign:
why are they doing
(I- P)(x)
what is the theoretical meenig of that
??