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Math Help - Span as small as possible

  1. #1
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    Span as small as possible

    Hello I do not indestand how I can solve this one, can someone give me a hint please. Thank you.
    the answer in the book is

    1 0
    0 1
    -1 0


    Find a subset of S with the same span as S that is as small as possible.

    1 -2 0
    0 0 1
    -1 2 0
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  2. #2
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    Re: Span as small as possible

    Column 2 is just a multiple of column one. Hence we can remove that vector (column 2) in the set S, and still have the same span, but with a smaller set.
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    Re: Span as small as possible

    Quote Originally Posted by mathproblems View Post
    Hello I do not indestand how I can solve this one, can someone give me a hint please. the answer in the book is
    1 0
    0 1
    -1 0

    Find a subset of S with the same span as S that is as small as possible.
    1 -2 0
    0 0 1
    -1 2 0
    \left[ {\begin{array}{*{20}c}   { - 2}  \\   0  \\   2  \\ \end{array} } \right] =  - 2\left[ {\begin{array}{*{20}c}   1  \\   0  \\   { - 1}  \\ \end{array} } \right] + 0\left[ {\begin{array}{*{20}c}   0  \\   1  \\   0  \\ \end{array} } \right]
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  4. #4
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    Re: Span as small as possible

    thank you!

    and the next problem is simular.
    1 -2 0
    -2 4 0
    1 -2 0

    it would be just
    1
    -2
    1

    because
    -2 *1 = -2
    -2*-2 = 4
    -2*1 = -2

    and also would be a reduced row echalon form need here?
    1 -2 0
    0 0 0
    0 0 0
    This is the way to find the -2 term in the correct way?
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  5. #5
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    Re: Span as small as possible

    and one more:
    -1 0 1
    0 1 2
    1 2 3

    Reduced row echalon form will be
    1 0 -1
    0 1 2
    0 0 0

    I still do not get this one...
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  6. #6
    fkf
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    Re: Span as small as possible

    Quote Originally Posted by mathproblems View Post
    thank you!

    and the next problem is simular.
    1 -2 0
    -2 4 0
    1 -2 0

    it would be just
    1
    -2
    1

    because
    -2 *1 = -2
    -2*-2 = 4
    -2*1 = -2

    and also would be a reduced row echalon form need here?
    1 -2 0
    0 0 0
    0 0 0
    This is the way to find the -2 term in the correct way?
    The 0 vector doesn't add anything new to the span since we can have the weight 0 on the first vector, and still get the 0 vector. Hence we can remove that and have the same span. The second column vector is still a multiple of the first one, and therefore not adding any new to the span so we can remove this one also, and hence only have the first column vector. Still having the same span as we had when we had the set S = {v1,v2,v3}

    The row echelon form is correct, and there you clearly we that the second column is just a multiple of the first column.
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  7. #7
    fkf
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    Re: Span as small as possible

    Quote Originally Posted by mathproblems View Post
    and one more:
    -1 0 1
    0 1 2
    1 2 3

    Reduced row echalon form will be
    1 0 -1
    0 1 2
    0 0 0

    I still do not get this one...
    Try to reduce the matrix once again, and you will hopefully have another answer.
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  8. #8
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    Re: Span as small as possible

    Not sure, how this can be reduced again?
    1 0 -1
    0 1 2
    0 0 0
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  9. #9
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    Re: Span as small as possible

    and one more:
    -1 0 1
    0 1 2
    1 2 3

    Reduced row echalon form will be
    1 0 -1
    0 1 2
    0 0 0


    the answer in the book is
    -1 0
    0 1
    1 2

    I still do not get this one...is it because the last column is
    -1
    2
    0 and it it bigger than other two
    1 0
    0 1
    0 0
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  10. #10
    fkf
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    Re: Span as small as possible

    Quote Originally Posted by mathproblems View Post
    and one more:
    -1 0 1
    0 1 2
    1 2 3

    Reduced row echalon form will be
    1 0 -1
    0 1 2
    0 0 0


    the answer in the book is
    -1 0
    0 1
    1 2

    I still do not get this one...is it because the last column is
    -1
    2
    0 and it it bigger than other two
    1 0
    0 1
    0 0
    The last column is just a linear combination of the first two columns, hence the last column doesn't add anything new to the span. Since it already can be "produced" (sorry for my English) by the other two column.
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  11. #11
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    Re: Span as small as possible

    thank you so much.
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