Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 0 down vote accepted

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.

share|cite|improve this answer
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.