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Math Help - Subspace

  1. #1
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    Subspace

    Let A= [1 -2 -3 0
    0 1 0 -1
    1 -1 1 0]

    and K={(x,y,z,w) E R^4: Av^t =0}

    Prove that K is a subspace of R^4. Hint: use a matrix equation rather than 3 linear equations.

    Any help is greatly appreciated!!
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  2. #2
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    Quote Originally Posted by Jacko View Post
    Let A= [1 -2 -3 0
    0 1 0 -1
    1 -1 1 0]

    and K={(x,y,z,w) E R^4: Av^t =0}

    Prove that K is a subspace of R^4. Hint: use a matrix equation rather than 3 linear equations.

    Any help is greatly appreciated!!
    See Linear subspace. K is a subspace because these three conditions hold:

    A0 = 0,

    Av = 0 \text{ and } Aw = 0 \text{ imply }A(v+w) = Av + Aw = 0,

    Av = 0 \text{ implies } A(av) = a(Av) = 0 \text{ for scalar } a.
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  3. #3
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    Yeah we've been told those conditions in lectures but only been shown basic examples.
    Anyone know how to do this question?
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  4. #4
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    Quote Originally Posted by Jacko View Post
    Yeah we've been told those conditions in lectures but only been shown basic examples.
    Anyone know how to do this question?

    Don't be scared of it, just show the conditions JakeD has mentioned.
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  5. #5
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    Quote Originally Posted by Jacko View Post
    Let A= [1 -2 -3 0
    0 1 0 -1
    1 -1 1 0]

    and K={(x,y,z,w) E R^4: Av^t =0}

    Prove that K is a subspace of R^4. Hint: use a matrix equation rather than 3 linear equations.

    Any help is greatly appreciated!!
    Quote Originally Posted by JakeD View Post
    See Linear subspace. K is a subspace because these three conditions hold:

    A0 = 0,

    Av = 0 \text{ and } Aw = 0 \text{ imply }A(v+w) = Av + Aw = 0,

    Av = 0 \text{ implies } A(av) = a(Av) = 0 \text{ for scalar } a.
    Quote Originally Posted by Jacko View Post
    Yeah we've been told those conditions in lectures but only been shown basic examples.
    Anyone know how to do this question?
    To show K is a subspace, you must show the three conditions listed in the linear subspace link hold.

    The first condition, for example, is 0 \in K. To show this holds, you note as I did that A0 = 0 so the vector 0 satisfies the equation defining K, which is Av = 0. Notice the hint says use a matrix equation.

    The other two conditions are done similarly.
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