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Linear algebra geometry of orthogonal decomposition

  1. Aug 17, 2010 #1
    1. The problem statement, all variables and given/known data
    y=[4 8 1]^T u_1 = [2/3 1/3 2/3]^T u_2=[-2/3 2/3 1/3]^T

    Part 1: Write y as the sum of a vector y hat in W and a vector z in W complement

    Part 2: Describe the geometric relationship between the plane W in R^3 and the vectors y hat and z from the part above.


    2. Relevant equations



    3. The attempt at a solution
    I got the first part I think.
    y = [2 4 5]^T + [2 4 -4]^T

    Part 2 is what i'm struggling with. I don't understand the relationship so I don't really know how to describe it. Can someone please help?
     
  2. jcsd
  3. Aug 17, 2010 #2
    Remember that a plane through the origin in [itex]\mathbb{R}^3[/itex] is completely determined by any line perpendicular to it. In other words, if you pick any line in 3-space, you're only going to be able to find one plane through the origin which is perpendicular to it.
     
  4. Aug 17, 2010 #3
    I'm still confused. The only thing I really know about this is that ý is called the orthogonal projection of y onto W. But I don't understand what each of the components of y actually mean or do.
     
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