1. ## othogonal projections - question about definition

I am a little bit confused by a wording on the orthogonal projection given in my study guide for linear algebra (LSE).

It says "it is the projection onto S parallel to the orthongonal complement of S (for a particular subspace of S)."

I am confused by the world 'parallel'. If we take a simple case of a plane S and its orthogonal complement So (a line perpendicular to the plane), then any vector x can be described as a sum of two projections: x=x1+x2, where x1 - is the projection of vector x onto a plane S, and x2 - the projection of vector x onto the line So. The projection of the vector onto a plane S - x1 - is not going to be parallel to the line So - since it will belong to the plane S, it will be perpendicular to So.

So, looking back at the definition, what is 'parallel' in my example?

thanks a lot! (sorry I cannot figure out how to cut and paste math signs into the message text).

2. Originally Posted by Volga
(sorry I cannot figure out how to cut and paste math signs into the message text).
Why cut and past when you can learn to use LaTex.

3. I think the narrative description was not too vague and someone can comment on the main question and not the side note in the parenthesis.

4. Originally Posted by Volga
So, looking back at the definition, what is 'parallel' in my example?
I understand. Choose for example in the usual euclidean vector space $\mathbb{R}^3$:

$S \equiv z=0$. Then $S_0 \equiv x=0 , y=0$

Choose $v=(1,1,1)$. We have the decomposition:

$(1,1,1)=(1,1,0)+(0,0,1),\quad x_1=(1,1,0)\in S,\;x_2=(0,0,1)\in S_0$

and $x_1$ is the orthogonal projection of $v$ onto $S$, and of course, $x_1$ is not parallel to $S_0$.

Now choose for example $P(2,3,5)$, the projection onto $S$ of the point $P$ parallel to the orthogonal complement of $S$ is the intersection of $P+\lambda (0,0,1)$ (parallel affine variety to $S_0$ through $P$) with $S$. In this case we obtain the point $(2,3,0)$

Fernando Revilla

5. Sorry Fernando it seems that your message got cut off in the end

6. Originally Posted by Volga
Sorry Fernando it seems that your message got cut off in the end
I don't see it. Anyway, ask just what you want.

Fernando Revilla

7. Fernando, I think Volga was referring to the fact that your last phrase in post # 4 is an incomplete sentence. At least, there's no verb there. It reads, "In this case the point (2,3,0)", which is incomplete.

8. Originally Posted by Ackbeet
Fernando, I think Volga was referring to the fact that your last phrase in post # 4 is an incomplete sentence. At least, there's no verb there. It reads, "In this case the point (2,3,0)", which is incomplete.
All right!. ( ). I've corrected it.

Thank you.

Fernando Revilla

9. You're welcome.

Cheers.

10. I think I got it. Projection is not a vector, it is a function (mapping) so by drawing a mapping line parallel to So we obtain (2,3,0) - projection of (2,3,5) onto S.

11. Originally Posted by Volga
... so by drawing a mapping line parallel to So we obtain (2,3,0) - projection of (2,3,5) onto S.
That's right.

Fernando Revilla