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Math Help - Finding a spanning set of a null space

  1. #1
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    Finding a spanning set of a null space

    Let U be the subspace of R4 given by:

    U = nullspace of the matrix


    [0 0 2 3 ]
    [0 -3 -2 -2 ]

    Find a spanning set for U.

    Any and all help will be awesome.

    Not sure how to start at all

    Cheers
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Mathmaticious View Post
    Let U be the subspace of R4 given by:

    U = nullspace of the matrix


    [0 0 2 3 ]
    [0 -3 -2 -2 ]

    Find a spanning set for U.

    Any and all help will be awesome.

    Not sure how to start at all

    Cheers
    The first thing to do is to find out what the nullspace of your matrix actually is - can you find a general form that these vectors take?

    Can you think of a spanning set for this vector space? For instance, if they were of the form \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \} then your space would be spanned by the vectors (2,0,0,0), (0,5,0,0), (0,0,6,0) and (0,0,0,1).

    I hope that that helps...
    Last edited by Swlabr; October 2nd 2009 at 08:19 AM. Reason: last vector in the basis was (0,0,0,d) instead of (0,0,0,1). So, naturally, I changed it.
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  3. #3
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    Quote Originally Posted by Swlabr View Post
    The first thing to do is to find out what the nullspace of your matrix actually is - can you find a general form that these vectors take?

    Can you think of a spanning set for this vector space? For instance, if they were of the form \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \} then your space would be spanned by the vectors (2,0,0,0), (0,5,0,0), (0,0,6,0) and (0,0,0,d).

    I hope that that helps...
    @Swlabr: Few questions please
    1. In your example \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \} what is the relevance of 2,5,6? Isn't it just same as \{a,b,c,d): a, b, c, d \in \mathbb{F} \}

    2. Nevertheless, is there a good/structured approach to find dimension and basis of a space given by something like \{a,2a,a+b,b): a, b \in \mathbb{F} \}
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by aman_cc View Post
    @Swlabr: Few questions please
    1. In your example \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \} what is the relevance of 2,5,6? Isn't it just same as \{a,b,c,d): a, b, c, d \in \mathbb{F} \}
    Yes. Yes it is. No excuses, other than to point that I was still correct...I mean, 4+2=80712/13452...

    2. Nevertheless, is there a good/structured approach to find dimension and basis of a space given by something like \{a,2a,a+b,b): a, b \in \mathbb{F} \}
    Yes. The answer to your problem is either one or two, but is clearly two as the entries in the 1st and 4th positions are independent of one another. It is no more than the number of variables given, but can be less if they cancel out with one another.

    (1, 2, 1, 0) and (0, 0, 1, 1) span the space you gave.
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  5. #5
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    Quote Originally Posted by Swlabr View Post
    Yes. Yes it is.



    Yes. The answer to your problem is either one or two, but is clearly two as the entries in the 1st and 4th positions are independent of one another. It is no more than the number of variables given, but can be less if they cancel out with one another.

    (a, 2a, a, 0) and (0, 0, b, b) span the space you gave.
    Thanks Swalbr
    If you won't mind can you explain why you say - "It is no more than the number of variables given"?
    Also is there is a formal way to solve such a problem - I am looking at something equivalent to row-reduction to find the rank of matrix. Thanks
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by aman_cc View Post
    Thanks Swalbr
    If you won't mind can you explain why you say - "It is no more than the number of variables given"?
    Also is there is a formal way to solve such a problem - I am looking at something equivalent to row-reduction to find the rank of matrix. Thanks
    In the vectors you gave you had an a and a b, so two variables. Say you had a subspace of vectors where you can write them like you did but with n>0 variables. Then you can easily split your general form into n vectors each with precisely one variable. In each vector you can take out this variable as a common factor, and the set of all of these vectors (the ones with the variables removed) forms a spanning set. You may be able to reduce this spanning set, if not then it is a basis.

    Does that make sense?

    Note also that a vector space cannot have a basis of order greater than the length of the vector...
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  7. #7
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    Thanks. I need to think this carefully though. I was also wondering if it has to do something with the fact that each component is a linear combination of the variables. What happens if we relax that - for e.g. {a^2,b^3,a+b^-1,b} (I have not checked if it is a sub-space.)
    Last edited by aman_cc; October 2nd 2009 at 08:46 AM.
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  8. #8
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    I presume you mean the matrix
    \begin{bmatrix}0 & 0 & 2 & 3 \\ 0 & -3 & -2 & -2\end{bmatrix}

    The kernel is, by definition, the set of vectors \begin{bmatrix}x \\ y \\ z \\ t\end{bmatrix} such that
    \begin{bmatrix}0 & 0 & 2 & 3 \\ 0 & -3 & -2 & -2\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ t\end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}

    which means we must have 2z+ 3t= 0 and -3y- 2z- 2t= 0. Those two equations drop the dimension from 4 to 2. We can write z= (-3/2)t and then y= (-1/3)(2z+ 2t)= (-1/3)(-3t+ 2t)= (1/3)t. That is, y and z depend on t while x, since it does not appear in the equations can be anything. Use x and t as "free variables". Any vector in the kernel are of the form \begin{bmatrix}x \\ (1/3)t \\ (-3/2)t \\ t\end{bmatrix}= x\begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix}+ t\begin{bmatrix}0 \\ 1/3 \\ -3/2 \\ 1\end{bmatrix}
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