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Vectors that form bases (linear algebra)

  1. Jan 2, 2008 #1
    [SOLVED] Vectors that form bases (linear algebra)

    1. The problem statement, all variables and given/known data
    I am given two vectors u and v, which are:

    u = (1/2 , 1/2 , 1/2 , 1/2) and
    v = (1/2 , 1/2 , -1/2 , -1/2).

    I have to find an orthonormal basis for R^4 containing u and v.

    3. The attempt at a solution
    The first thing that came to me was Gram-Schmidt - but then I saw that the dot-product between u and v is zero, so Gram-Schmidt is overkill.

    I just need to find two other linearly independant vectors that has dot-product equal zero with respectively u and v. Is that even possible and how would I do that?
     
  2. jcsd
  3. Jan 2, 2008 #2

    HallsofIvy

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    Staff Emeritus
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    Okay, u and v are already orthogonal so you really just need to find two more vectors that are orthogonal to both u and v (and to each other). Let w= (a, b, c, d). Then [itex]u\cdot w[/itex] is (1/2)a+ (1/2)b+ (1/2)c+ (1/2)d and you want that equal to 0. That, of course, is the same as a+ b+ c+ d= 0. [itex]v\cdot w[/itex] is (1/2)a+ (1/2)b- (1/2)c- (1/2)d= 0 which is the same as a+ b- c- d= 0. Adding the two equation, you get 2a+ 2b= 0 or b= -a. With b= -a, the first equation becomes c+ d= 0 or c= -d. Choose a and d to be whatever you want and you will get a vector orthogonal to both u and v. You should be able to choose one value of a and d to get a vector orthogonal to u and v, then choose another value of a and d to get a vector that is orthogonal to u and v and also to your first vector.
     
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