Does zero vector only exist in dim 0?

  • Thread starter pob1212
  • Start date
  • #1
21
0
Hi,

just curious, is it false to say the vector 0 exists in R^n, n>0 and a natural number?
i.e. does x = [0, 0, 0, 0] exist in R^4, or simply the zero space?
My guess is that the first statement is not false given that the zero space is a subspace of R^n, n>0.

Am I right or wrong?

Thanks
 

Answers and Replies

  • #2
806
23
The zero vector is the additive identity; it exists in all Rn, and is defined as [itex]<0, 0, ... , 0>[/itex].

EDIT: My statement is misleading. "The zero vector" does not exist in every Rn; rather, for every n, there exists an element of Rn which is the additive identity and is called the zero vector.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
961
But in what sense is the vector "which is the additive identity and is called the zero vector" NOT the "zero vector"?

But, having written that, I do see your point. It is a mistake to think that there exists a single vector that is in all vector spaces. In, for example, R3, the "zero vector" is (0, 0, 0). But in the vector space of "all polynomials of degree 2 or less" the "zero vector" is the function f(x) such that f(x)= 0.

Every vector space contains a zero vector but it is misleading to talk about "the" zero vector as if all zero vectors were the same thing.

(But all vector spaces of dimension n are isomorphic and that isomorphism maps the zero vector in one to the zero vector in the other. A "natural isomorphism" from R3 to the space of polynomials of degree 2 or less takes (a, b, c) to ax^2+ bx+ c and so maps (0, 0, 0) to 0x^2+ 0x+ 0= 0 for all x.)
 

Related Threads on Does zero vector only exist in dim 0?

Replies
4
Views
1K
  • Last Post
Replies
17
Views
1K
Replies
7
Views
877
  • Last Post
Replies
15
Views
6K
  • Last Post
Replies
4
Views
21K
  • Last Post
Replies
18
Views
17K
Replies
9
Views
943
  • Last Post
Replies
8
Views
3K
Replies
8
Views
3K
  • Last Post
Replies
3
Views
1K
Top