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Homework Help: 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


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    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


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    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


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    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


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    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


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    Could you post the problem exactly as given to you? It seems like you're leaving a lot of relevant details out.
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