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Math Help - Vectors span R^3

  1. #1
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    Vectors span R^3

    Which of the following sets of vectors span R^3?
    a. {(1,-1,2),(0,1,1)}
    b. {(1,2,-1),(6,3,0),(4,-1,2),(2,-5,4)}
    c. {(2,2,3),(-1,-2,1),(0,1,0)}
    d. {(1,0,0),(0,1,0),(0,0,1),(1,1,1)}
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  2. #2
    Member Ruun's Avatar
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    Hi!

    I'm spanish so I hope that span means... ehm "make" \mathbb{R}^3. If you have a vector space for example \mathbb{R}^n, then the dimension is n. The dimension of a vector, as far as I know, it's a maximal set of linear independent vectors or a minimal set of "generator" vectors. In practise it's the number of vector that every base should have. In our case as n=3 then the dimension is 3.

    With this in mind, option 1) doesn't have 3 vectors so it can't span \mathbb{R}^3. 2) Four vectors, it can be a good candidate. If there are 3 or more linearly independent then it's ok. This is the math around this.

    To proof if they are or not linearly independent, just look if there is a determinant of a matrix that have as rows or columns your vectors, different of zero.

    Hope it helps
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  3. #3
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    I tried to follow examples of how to solve this problem in my textbook, but none of them were in the same format.

    For part a, my work is:










    Reduced


    The system is inconsistent and has no solution so this set does not span R^3?

    Part b: same work resulting in the following reduced matrix


    So this would also not span R^3 since there are no solutions?

    Part c:




    So this set does span R^3?

    Does anyone know if this is correct? Thank you!
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  4. #4
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    You posted this same thing under "urgent homework help" and I responded there. Didn't you notice where it said "Don't make a duplicate thread elsewhere, please"?
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