1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Dimensions of subspaces

  1. Oct 30, 2008 #1
    If V and W are 2-dimensional subspaces of R4 , what are the possible dimensions of the subspace V intersection W?

    I am new to subspaces, so I have no clue to this question. Help guys!!!

    Options: (A) 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, and 2 only (E) 0, 1, 2, 3, and 4
     
  2. jcsd
  3. Oct 30, 2008 #2

    Mark44

    Staff: Mentor

    You don't need to be concerned about the fact that V and W are subspaces, nor that they are embedded in a four-dimensional space.

    V and W are two-dimensional, so you know what they look like geometrically, don't you? Now, what are all of the possibilities for the intersection of V and W?
     
  4. Oct 30, 2008 #3
    you mean to say that V and W are 2x2 matrices and their intersection would also be a 2x2 matrix....so the required dimension is 2 only?
     
  5. Oct 30, 2008 #4

    Mark44

    Staff: Mentor

    No, I didn't say that at all. What does any two-dimensional space look like geometrically?
     
  6. Oct 30, 2008 #5
    I am sorry but i am not able to grasp the idea u r trying to put.
     
  7. Oct 30, 2008 #6
    I think the possibility of dimensions would be 0,1, and 2.
     
  8. Oct 30, 2008 #7

    Mark44

    Staff: Mentor

    Considering the real plane, what is the dimension of a single point? What is the dimension of a line?
     
  9. Oct 30, 2008 #8
    it is single dimension
     
  10. Oct 30, 2008 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You do need to pay some attention to the dimension of the space they are embedded in. If they were embedded in three space a 0 dimensional intersection would be impossible.
     
  11. Oct 30, 2008 #10

    Mark44

    Staff: Mentor

    I asked two questions. Your answer above is the answer to which question?
     
  12. Oct 30, 2008 #11

    Mark44

    Staff: Mentor

    Yes, but I wanted the OP to focus more on the subspaces and not get tangled up in trying to imagine a 4-D space.
     
  13. Oct 31, 2008 #12
    point is 0 dimension and line is single dimension
     
  14. Oct 31, 2008 #13

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    And a plane has 2 dimensions.
     
  15. Oct 31, 2008 #14

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You also need to pay attention to the fact that they are subspaces. The planes (x, y, 0, 0) and (x, y, 0 , 1) have empty intersection. That is impossible for subspaces.
     
  16. Oct 31, 2008 #15
    I got the answer of my question, really appreciate ur posts for making it clear to me. Thnx a bunch!!

    why is it impossible for subspaces?
     
  17. Oct 31, 2008 #16
    How come a point can exist as an intersection of subspaces in 4 space but not in 3 space?
     
  18. Oct 31, 2008 #17

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The 2-dimensional subspaces of R4, {(x, y, 0, 0)} and {0, 0, u, v}, have only the single point (0, 0, 0, 0) in common. Two 2- dimensional subspaces in R2 must have a least a 1-dimensional subspace in common.
     
  19. Oct 31, 2008 #18

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If the intersection is the single point 0, then if you take union of a basis for one subspace with a basis of the other, the whole set must be linearly independent (otherwise you could construct another intersection point). In the case of two planes, that's 4 linearly independent vectors. Possible in 4 space, not in 3 space.
     
  20. Oct 31, 2008 #19
    those were the new concepts for me....thnx

    How do I proceed for such type of questions:

    Let V be the real vector space of all real 2 x 3 matrices, and let W be the real vector space of all real 4 x 1
    column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace
    {v belongs to V:T(v)=0}?
     
  21. Oct 31, 2008 #20

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    In other words, "what is the dimension of the kernel of T" or "what is the nullity of T".
    Is that really the entire question? There are many possible answers. For example, T(v)= 0 for all v is a linear transformation and it's kernel is all of V. Since V is a 6 dimensional space (Taking Vij to be the matrix with 1 at "ith row, jth column" and 0 everywhere else, for i= 1 to 2, j= 1 to 3 gives a basis), so that in this example, the dimension of the kernel would be 6.

    On the other hand the linear transformation that maps
    [tex]\left[\begin{array}{ccc} a & b & c \\d & e & f\end{array}\right]\right)[/tex] to [tex]\left[\begin{array}{c} a \\ 0 \\ 0 \\ 0\end{array}\right][/tex]
    has nullity 5: dimension of the kernel is 5.

    Perhaps the problem is "what is the smallest possible dimension of the kernel of T".
    Since the largest the image of T can be is 4 dimensional, the dimension of "the real vector space of all real 4 x 1 column vectors", the smallest the kernel can be is 6- 4= 2.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Dimensions of subspaces
  1. Subspaces & dimension (Replies: 9)

  2. Dimension of subspace (Replies: 8)

Loading...