1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Orthogonal Complements

  1. May 8, 2013 #1
    1. The problem statement, all variables and given/known data

    Let L: ℝn→ℝm be a linear transformation with matrix A ( with respect to the standard basis). Show that ker(L) is the orthogonal complement of the row space of A.

    2. Relevant equations



    3. The attempt at a solution
    The ker(L) is the subset of all vectors of V that map to 0. The orthogonal complement of W is the set of all vectors x with property that xw= 0. Would we use the dimension theorem:
    dim(ker(L)) = n - dim(range(L)) = n - dim(W) =dim(Wτ). Since Wτ is contained in ker(L).

    *Wτ is orthogonal complement.
     
  2. jcsd
  3. May 8, 2013 #2
    This seems way to obvious. The row space is the subset of all vectors that are linear combinations of rows of A. So the kernel of L is the subset of all vectors in V that map to 0. Then, the orthogonal complement is the set of vectors with the property that x ⋅ w = 0. So, I need to prove that the vector x in the orthogonal complement is the same vector in the row space?
     
  4. May 8, 2013 #3

    Mark44

    Staff: Mentor

    Let's use W in a meaningful way since you haven't defined what W represents. Let W = ker(L).

    The orthogonal complement of W is the set of all vectors x with property that x ##\cdot## w= 0, where w ##\in## W.

    Now, assume that x is any vector in the row space of L. How can you write x, knowing what you know about L and its matrix representation?
     
  5. May 9, 2013 #4
    If a is scalars, then ax1 + ax2 + ..... + axk = 0
     
  6. May 9, 2013 #5

    Mark44

    Staff: Mentor

    What does this equation mean? I have no idea what you're doing.
     
  7. May 9, 2013 #6
    That was the linear combination of the row space. That was how I thought to write x in terms of L. And it's equal to 0 because it is the kernel.
     
  8. May 9, 2013 #7

    Mark44

    Staff: Mentor

    There are several things wrong with this.

    1. There are not k rows in the matrix, so it makes no sense to list them as x1, x2, ... , xk.
    2. The row space is not in the kernel. Your goal in this problem is to show that the row space is the orthogonal complement of the kernel.
    3. A vector in the kernel doesn't have to be 0.
     
  9. May 9, 2013 #8
    Okay, given matrix A and vector x:

    A⋅x = 0 means that wkx = 0 for ever row vector wk in R. Therefore, the orthogonal complement of row space is kernel.
     
  10. May 9, 2013 #9

    Mark44

    Staff: Mentor

    Any old vector x? Is x in the row space of A or is it in the kernel of A?
    I don't follow this. Ax = 0 means only that x is in the kernel of A. Why does it follow that wkx = 0? Neither wk or x has to be the zero vector.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted