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Thread: spanning set for R^2

  1. #1
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    Question spanning set for R^2

    First, can anyone explain to me the meaning of spanning set? I am a little confused with it meaning.

    Then please show me how to do the below questions. Thank you very much.

    Determine whether any of the following sets are spanning sets for R^2 , considered as column matrices
    1) S = { esub1 = [1] , esub2 = [0]}
    ........................[0]............. [1]

    2) S = { esub1 = [1] , esub2 = [0], fsub1 = [1]}
    ........................[0]............. [1].............[1]

    3) S = { fsub1 = [1] , fsub2 = [2]}
    .......................[1]............. [2]


    =================
    note: e and f are vectors.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by apple12 View Post
    First, can anyone explain to me the meaning of spanning set? I am a little confused with it meaning.
    A spanning set for a (sub-) space $\displaystyle W$ is a set of elements of $\displaystyle W$ which spans $\displaystyle W$. A set $\displaystyle S \subset W$ spans $\displaystyle W$ if for all $\displaystyle w \in W$ $\displaystyle w$ is a linear combination of elements of $\displaystyle S$.

    Then please show me how to do the below questions. Thank you very much.

    Determine whether any of the following sets are spanning sets for R^2 , considered as column matrices
    1) S = { esub1 = [1] , esub2 = [0]}
    ........................[0]............. [1]
    This is the standard basis for $\displaystyle R^2$, so spans $\displaystyle R^2$.

    2) S = { esub1 = [1] , esub2 = [0], fsub1 = [1]}
    ........................[0]............. [1].............[1]
    This contains the standard basis and so spans $\displaystyle R^2$

    3) S = { fsub1 = [1] , fsub2 = [2]}
    .......................[1]............. [2]
    $\displaystyle [1,2]'$ cannot be written as a linear combination of elements of $\displaystyle S$, so $\displaystyle S$ does not span $\displaystyle R^2$.

    RonL
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