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Is R^n Euclidean Space a vector space too?

  1. Sep 4, 2015 #1
    Dear Physics Forum personnel,

    I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n?

    Sincerely,


    PK
     
  2. jcsd
  3. Sep 4, 2015 #2

    micromass

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    Yes.

    No clue what you mean.
     
  4. Sep 4, 2015 #3
    As for the second question, I mean if the 1x2 matrix (a1, a2) or its 2x1 form (column vector) can be considered as a vector for the R^2 since the R^2 is basically the collection of real numbers in the ordered pair (a, b)?
     
  5. Sep 4, 2015 #4

    micromass

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    There is a linear isomorphism between the ##1\times 2##-matrices and ##\mathbb{R}^2##, yes.
     
  6. Sep 4, 2015 #5

    HallsofIvy

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    The Eulclidean space Rn is geometric- there are such things as points and distances defined but no "operations". A vector space is algebraic we must have operations such as sum and scalar multiplication defined. Of course, for, finite dimensional Rn, we can define the sum as (x1, x2,... , xn)+ (y1, y2, ..., yn)= (x1+ y1, x2+ y2... , xn+ yn) and scalar multiplication defined as a(x1, x2, ..., xn)= (ax1, ax2, ..., axn). If we consider those operations as "natural" then we can think of this as a "natural" correspondence between Rn and an n dimensional vector space.[/sub][/sub]
     
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