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Homework Help: Subspaces and Basis

  1. Nov 8, 2008 #1
    1. The problem statement, all variables and given/known data

    Let u be a vector where u = [4 3 1]. Let A be the set of all vectors orthogonal to u. Show that A is subspace of R^3. Then find the basis for A.

    2. Relevant equations

    3. The attempt at a solution

    For showing that A is a subspace...

    Zero vector is in A because A(0) = 0

    For any u & v, u+v is in A because Au=0, Av=0, and A(u+v) = Au+Av = 0

    And for any scalar c, A(cu) = c(Au) = c(o) = 0

    As for the basis, I really have no idea where to even start with that.

    Thanks for any help.
  2. jcsd
  3. Nov 8, 2008 #2


    Staff: Mentor

    You should do the first part of this problem; namely, finding the set of vectors that are orthogonal to u = (4, 3, 1). How can you tell that an arbitrary vector (x, y, z) is orthogonal to a given vector?
    None of this makes any sense. A is a set, not a matrix, so it doesn't make any sense to multiply a vector by A.

    As for finding a basis for A, if you do the first part you will be on your way toward a basis.
  4. Nov 9, 2008 #3
    I'm not entirely sure how to show an arbitrary vector is orthogonal to a given vector. I've looked through my text for help, but it's not really helping.
  5. Nov 9, 2008 #4


    Staff: Mentor

    Is "dot product" a hint?
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