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In Apostol's Calculus Volume II, he introduces ten axioms that define a linear space. Thereafter, an example (example 4, page 5) is given of a linear space. This fourth example I do not understand. It states:

 EXAMPLE 4. Let V be the set of all vectors in Vn, orthogonal to a given nonzero vector
   N. If n = 2, this linear space is a line through 0 with N as a normal vector. If 
   n=3, it is a plane through 0 with N as normal vector.

The reason for my difficulty in understanding this example is due to the fact that I cannot visualize it. I understand (correct me if I'm wrong), that when saying n=2, we are saying that there are two dimensions and when n=3, there are three. This is why there is line in the first case, and a plane in the second. Now, more specifically my questions are:

 -Is there a physical (or 'everyday' normal) example or analogy of this case above? 
 -why is the line passing only through zero? why not anywhere else? 

I am just beginning a long journey through analytic math with books like Apostol's. Any tips for such an endeavor would be wonderful as well.

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2 Answers 2

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Let's start with

-"why is the line passing only through zero? why not anywhere else?"

This is because all subspaces must share the zero vector with the parent space.

-"Is there a physical (or 'everyday' normal) example or analogy of this case above?"

Sure. Let $S$ be a differentiable surface in 3 dimensions. Then the vector space of tangent vectors at a given point is exactly the space of vectors orthogonal to the normal vector of the surface at that point.

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Ah, thanks. Your first point is in accord with the fifth axiom then too, Existence of a Non-Zero element. –  user1711997 Nov 13 '12 at 19:57

For $n=2$, think of the usual plane. Let the vector be $(1,1)$, along the line $x=y$. Then the space orthogonal to it is the line perpendicular, the line $x=-y$. You can verify that any vector along that line is orthogonal to $(1,1)$

For $n=3$ think of usual 3-space. Let the vector be $(1,1,0)$, then the orthogonal space is the plane containing $x=-y$ and the $z$ axis.

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