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Math Help - Span of R^3

  1. #1
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    Span of R^3

    I am having problems with this question could anyone help me?

    Let u={u1,u2,u3}, v={v1,v2,v3} w={w1,w2,w3} be three vectors in R^3. Show that S={u,v,w} spans R^3 if an only if determinent of

    u1 u2 u3
    v1 v2 v3
    w1 w2 w3

    does not equal zero.

    EDIT:

    here is my attempt does it sound solid?

    Since the determinent does not equal zero then the matrix is row equivalent to the Identity matrix. Then the augmented matrix

    u1 u2 u3 a
    v1 v2 v3 b
    w1 w2 w3 c

    can be written in reduced row echelon form and therefore S spans R^3
    Last edited by mulaosmanovicben; March 7th 2010 at 11:18 AM.
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  2. #2
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    Quote Originally Posted by mulaosmanovicben View Post
    I am having problems with this question could anyone help me?

    Let u={u1,u2,u3}, v={v1,v2,v3} w={w1,w2,w3} be three vectors in R^3. Show that S={u,v,w} spans R^3 if an only if determinent of

    u1 u2 u3
    v1 v2 v3
    w1 w2 w3

    does not equal zero.

    EDIT:

    here is my attempt does it sound solid?

    Since the determinent does not equal zero then the matrix is row equivalent to the Identity matrix. Then the augmented matrix

    u1 u2 u3 a
    v1 v2 v3 b
    w1 w2 w3 c

    can be written in reduced row echelon form and therefore S spans R^3
    If the det=0, then the vectors are lin. dep.

    1 vector in \mathbb{R}^3 forms a line
    2 vectors in \mathbb{R}^3 at most forms a plane

    2 lin. ind. vectors can only span \mathbb{R}^2
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