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Subspace of 2D space of physics

  1. Feb 25, 2014 #1
    Please excuse me for my less knowledge. I always tried to physically visualise mathematics facts.
    My first question is " Is 1D space of physics a subspace of 2D space of physics and so on......
    So in this way our 3D space is a subspace of 4 D space(spacetime).
    Can I imagine applying all properties of vector space applicable in physical world?
    My study of Advanced Algebra is still in infancy.
    Please let me know if my question is irrelevant.
    Thank you all in advance.
    Please reply so that I start further discussion related to Advanced Algebra. I want to master it.

    In classical mechanics some process can be defined in terms of generalised coordinates like density , temperature , location , time and so on....
    Now a space is defined with independent dimensions density,temperature,location, time and suppose colour.
    Is space generated by density, temperature and location a subspace of above space.
    Is it embedded in the above space?
    Can I have two different subspaces of five dimensions of the above space?
     
    Last edited: Feb 25, 2014
  2. jcsd
  3. Feb 25, 2014 #2

    mathman

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    If n < m, then n dimensional space can be considered a subspace of m dimensional space.
    Example: Let{(x,y,z)|x,y,z real} be a 3 dimensional space. Then {(x,y,a)|x,y real, a fixed} is a 2 dimensional subspace for each value of a.

    I think this should help you with the second question. You can have as many subspaces as you want.
     
  4. Feb 25, 2014 #3

    HallsofIvy

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    Notice the difference between your question "Is 1D space of physics a subspace of 2D space of physics and so on...... So in this way our 3D space is a subspace of 4 D space(spacetime)."

    and mathman's response "If n < m, then n dimensional space can be considered a subspace of m dimensional space."

    Strictly speaking, no, 1D space is NOT a subspace of 2D space and 3D space is not a subspace of 4Dspace. Points in 1D space can be designated by a single number, a, while points in 2D space are designated by pairs of numbers, (x, y). But we can associate the point, a, with the pair (a, 0) so there is an "isomorphism" between 1D and a subspace of 2D. This is NOT the same as saying 1D is a subspace itself because there are many different such "isomorphisms" or assignments: a with (0, a) or with (a, a) or (a, ma) for fixed m, etc.
     
  5. Mar 5, 2014 #4
    Thank you dear friends!
     
  6. Mar 9, 2014 #5
    To a large extent physics in the plane can be considered a super position of physics on two lines. However, be careful. Special relativity with physics in 4 D (Space-time) is like physics in conventional four dimensions except the metric is different. The interval in special relativity is (often): sqr root(x squared + y squared + z squared - time squared).
     
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