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Linear Mappings Question

  1. Jan 11, 2010 #1
    1. The problem statement, all variables and given/known data

    http://img526.imageshack.us/img526/743/93134049.png [Broken]


    2. Relevant equations
    Just the standard linear mapping properties and theorums.


    3. The attempt at a solution

    I have already solved part A by considering the transformation A: Rn -> R | A(x) = a.x where x is a vector in Rn, and finding the dimension of the nullspace.

    I am stuck on where to begin the proof for part b, once I have b I don't think I will have a problem generalizing it for part c.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jan 11, 2010 #2
    Your solution to part (a) already contains the essence of a solution to part (b); just think about the dimension of the null space of a different linear map [tex]B[/tex], one that uses the existence of [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] both.

    (If you need a further hint: given that you are supposed to find that [tex]\dim(S \cap T) = n - 2[/tex] in case [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] are linearly independent, what do you think the dimension of the range space of [tex]B[/tex] ought to be?)
     
  4. Jan 11, 2010 #3
    ah, thank you, very good. I have a transformation B: Rn -> R2 that works nicely, and the generalization follows quite easily.
     
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