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!

Trying to understand(conceptually) orthogonality

  1. Mar 21, 2016 #1
    1. The problem statement, all variables and given/known data
    if Ax = b has a solution and A^Ty = 0 , is y^T(x) = 0 or y^T(y) =0

    2. Relevant equations


    3. The attempt at a solution
    I simply do not think i understand the properties to answer this question.

    From my understandinging, the transpose of A times y is = 0. This means that A transpose and y(all the members of these two subspaces) are perpendicular. Does this indicate that y is essentially the nullspace of A transpose? A transpose is the rowspace?

    Where do I progress on frmo this?

    I'd like to add... why is the transpose of these matrices even relevant? I don't understand. Are there properties of the transpose that I'm missing? What are these variables y and x representing? There's just so much here that's confusing me(I'm guessing I don't understand fundamentally)

    I know that the row space is orthogonal to the nullspace(and colspace with the left nullspace) how do i use this knowledge to solve this problem?
     
    Last edited: Mar 21, 2016
  2. jcsd
  3. Mar 21, 2016 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Do you mean Ax=b has a unique solution?
    If it has a unique solution then A is invertible, and hence so is ##A^T##, which allows us to choose between the two alternative answers.

    But if we only know that Ax=b has at least one solution, then A may be singular, in which case the question is unanswerable.
     
  4. Mar 21, 2016 #3
    Yes it has a unique solution.

    What good does it knowing that A^T and ax=b is invertible?
     
  5. Mar 21, 2016 #4

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You are given the equation ##A^Ty=0##. Using the fact that ##A## is invertible, and hence so is ##A^T##, rearrange that equation so that y is alone on one side (ie make y the subject of the equation). Then simplify as much as possible.
     
  6. Mar 21, 2016 #5
    Well if I were to isolate y from that equation you've stated, wouldn't it just be y = 0? the fact that A is invertible(meaning so is its transpose) confuses me. Why does that help me?
     
  7. Mar 21, 2016 #6

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    They've asked you if ##y^Tx = 0## or ##y^Ty =0##. The answer is yes if either or both is true.
    If you can show that ##y=0##, can you then prove that at least one of those two is true?

    If so, the answer to the set question is Yes.
     
  8. Mar 21, 2016 #7

    Mark44

    Staff: Mentor

    What do y^T(x) and y^T(y) mean?
    It would be helpful to state that x and y are vectors in some vector space (such as ##\mathbb{R}^n##?).

    Since your question is about orthoganality, there's an inner product lurking about somewhere, so presumably y^T(x) could just as well be written as ##\vec{y} \cdot \vec{x}## and y^T(y) could be written as ##\vec{y} \cdot \vec{y}##
    That doesn't make any sense. Two vectors can be perpendicular (or othogonal), but a matrix and a vector aren't things that can be perpendicular to each other.
    If ATy = 0, then, yes, y is in the nullspace of AT.
     
  9. Mar 21, 2016 #8

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Could you post the problem exactly as given to you? It seems like you're leaving a lot of relevant details out.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Trying to understand(conceptually) orthogonality
Loading...