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Math Help - Find a basis for the subspace S of R3 spanned by....

  1. #1
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    Unhappy Find a basis for the subspace S of R3 spanned by....

    Hi, i am struggling with the idea of basis and dim.



    Find a basis for the subspace S of R3 spanned by
    { v1 = (1,2,2), v2 = (3,2,1), v3 = (11,10,7), v4 = (7,6,4) }.
    What is dim s ?
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  2. #2
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    A basis is a spanning set that is also independent. You are given that this subspace is spanned by these four vectors so a basis would be a subset of that. Start with any one of them, say, v1.
    v2 is not a multiple of v1 so {v1, v2} are independent.

    Now look at v3. Is v3 independent of v1 and v2? That is, can v3 be written as a linear combination of v1 and v2? That is the same as asking "can we have av1+ bv2+ cv3= 0 where a, b, c are not all 0?".
    If yes, add v3. If no, ask the same about v4.

    Since R^3 itself has dimension 3, a basis cannot contain more than 3 vectors.
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  3. #3
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    ok, so i got

    v3 = 2(v1) + 3(v2)
    v4 = v1 + 2(v2)

    so the basis is v1 and v2 and dimension 2?

    Thanks for your help
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  4. #4
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    Yes, well done!
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