Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Does zero vector only exist in dim 0?

  1. Sep 24, 2012 #1
    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
     
  2. jcsd
  3. Sep 24, 2012 #2
    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.
     
  4. Sep 25, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Does zero vector only exist in dim 0?
  1. Zero Vector (Replies: 15)

  2. The Zero Vector (Replies: 4)

  3. Does 0^0 = 1 (Replies: 106)

Loading...