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A linear algebra question

  1. Aug 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Is a set of n-tuples which must respect the conditions of closure under addition and closure under scalar multiplication a vector space or a vector subspace?

    That is, in a 3-dimensional space, are planes which pass by the origin considered to be subspaces of the 3-dimensinal space in question? Or are they considered to be vector spaces?

    The place where I was reading about it said that subspace of R n and Euclidean vector space are the same thing, but I'm not sure whether it's true or not. I probably misunderstood something.

    2. Relevant equations
    -

    3. The attempt at a solution
    Posting here, as well as Googling.
     
  2. jcsd
  3. Aug 7, 2012 #2
    A subspace is always space. What makes a space a subspace is that it is "immersed" in some bigger space. An Euclidean plane is a vector space in its own right, but a subspace in 3D space, etc.
     
  4. Aug 7, 2012 #3

    Ray Vickson

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    A two-dimensional plane in a 3-dimensional space is not, itself, a subspace unless it passes through the origin. If it misses the origin entirely, then closure under addition and/or multplication by a scalar fails.

    RGV
     
  5. Aug 7, 2012 #4
    Thanks for your help.

    I have another question: Would it be correct to say that R3 and R2 are both vector spaces?
     
    Last edited: Aug 7, 2012
  6. Aug 8, 2012 #5
    Yes, R^3 and R^2 are both vector spaces.

    R^n for every positive integer is also a vector space.

    Check out the Linear Algebra wikibook, and read up on the chapter on Vector Spaces.
    http://en.wikibooks.org/wiki/Linear_Algebra
     
  7. Aug 9, 2012 #6
    Subspace Rn is an Euclidean vector space iff, in addition to satisfying closure under addition and multiplication with the inclusion of zero element, its structure (inner product) is defined as dot product.
     
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