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Homework Help: Linear Algebra, subspaces

  1. Jun 11, 2008 #1
    There are more answers to this problem, but I'm not sure how to approach it.

    The subspaces of [tex]R^3[/tex] are planes, lines, [tex]R^3[/tex] itself, or Z containing only (0,0,0,0).

    b. Describe the five type of subspaces of [tex]R^4[/tex]

    i. lines thru (0,0,0,0)
    ii. zero (0,0,0,0)

    iii. planes thru (0,0,0,0)
    What do you mean by planes? Are you assuming 2 dimensional sets as is usual with planes?

    iv. itself, [tex]R^4[/tex]

    What's the 5th?
     
    Last edited by a moderator: Jun 11, 2008
  2. jcsd
  3. Jun 11, 2008 #2

    HallsofIvy

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    There are more answers to this problem, but I'm not sure how to approach it.

    The subspaces of [tex]R^3[/tex] are planes[/quote]
    Planes containing the origin
    Lines through the origin
    caution: Z is typically used to mean the set of integers. Here, I guess you mean "the set containing only (0, 0, 0)". (NOT (0,0,0,0)!)

    More correctly, the set containing only the zero vector.

    What do you mean by planes? Are you assuming 2 dimensional sets as is usual with planes?

    Look at number iii again. You have incorporated two different "kinds" of subspace there.
    (R4 is 4 dimensional. Think of your five kinds of subspace as: 0, 1, 2, 3, 4.)
     
  4. Jun 14, 2008 #3
    So I have planes in R^4. But I also have planes in R^2 and R^3 passing thru (0,0,0,0) that are contained in R^4. Lastly, R^4, itself.
     
  5. Jun 15, 2008 #4

    matt grime

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    It doesn't make sense to talk of a plane in R^3 being contained in R^4 as if that were a well defined object. Even if we identify a copy of R^3 sitting inside R^4, any plane in R^3 is still a plane in R^4 so you just counted it when you specified the set of planes (passing through the origin). Also, R^2 contains only one 2-dimensional subspace - itself.

    Doing it in coords, isn't the set of things (x,y,z,0) a subspace of R^4? What is its dimension?
     
  6. Jun 15, 2008 #5

    HallsofIvy

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    Once again what do you mean by "planes"? How is a "plane in R^2" different from a "plane in R^3" or a "plane in R^4"?
     
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