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Proofs of subspaces in R^n (intersection, sums, etc.)

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Let E and F be two subspaces of R^n. Prove the following statements:

    (n means "intersection")
    1. If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors
      Note: Above zero denotes the zero vector in R^n
    2. EnF = {u, such that u is in E, and u is in F} is a subspace of R^n
    3. E+F = {w=u+v, u is in E, v is in F} is a subspace of R^n
    4. If EnF={0} then dim(E+F)=dim(E)+dim(F)
  2. jcsd
  3. Nov 16, 2009 #2


    Staff: Mentor

    If you want some help (which is probably why you're here), you need to show what you have tried to do.
  4. Nov 17, 2009 #3
    for the intersection questions, think about closure under addition (and subtraction)
    for the dimension question, think about what would happen if vectors "overlapped" in two spaces..
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