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Math Help - Justify basis

  1. #1
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    Justify basis

    Suppose that {v1, v2, v3, v4} is a basis for a subspace V of  R^n. Which, if any, of the following subsets are bases for V?
    1) S1 = {v1 + v2, v2 + v3, v3 + v4, v4 + v1}
    2) S2 = {v1, v1 + v2, v1 + v2 + v3, v1 + v2 + v3 + v4}
    3) S3 = {v1 - v2, v2 - v3, v3 - v4, v4 - v1}
    4) S4 = {v1 + v2, v2+ v3, v3 + v4}
    5) S5 = {v1, v1 + v2, v2 + v3, v3 + v4, v1 + v2 + v3 + v4}
    Justify your answers?

    Can someone just show me how to do one of them, then I wil be able to do rest of them?
    I know a basis needs to be a span of V and linearly independent, I am not sure how to prove it is a span of V
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  2. #2
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by 450081592 View Post
    Suppose that {v1, v2, v3, v4} is a basis for a subspace V of  R^n. Which, if any, of the following subsets are bases for V?
    1) S1 = {v1 + v2, v2 + v3, v3 + v4, v4 + v1}
    2) S2 = {v1, v1 + v2, v1 + v2 + v3, v1 + v2 + v3 + v4}
    3) S3 = {v1 - v2, v2 - v3, v3 - v4, v4 - v1}
    4) S4 = {v1 + v2, v2+ v3, v3 + v4}
    5) S5 = {v1, v1 + v2, v2 + v3, v3 + v4, v1 + v2 + v3 + v4}
    Justify your answers?

    Can someone just show me how to do one of them, then I wil be able to do rest of them?
    I know a basis needs to be a span of V and linearly independent, I am not sure how to prove it is a span of V
    Ok good.
    I think that to prove that S1 a span of V, if you can show that v1, v2, v3 and v4 can be written as a linear combination of the vectors forming S1, it means it spans the same vector space, S1 in this case.
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  3. #3
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    Smile

    hi
    S_5 is obviously not a basis of V, since,
    (v_1 + v_2 + v_3 + v_4)=(v_1+v_2)+(v_3+v_4).
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  4. #4
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    Smile

    S_1 is also not a basis since,
    v_4+v_1=(v_4+v_3)+(v_1+v_2)-(v_2+v_3).
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  5. #5
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    We can note from the given base that dim V = 4 and therefore:

    S_5 is not a base since |S_5| = 5 > 4 = |\{v_1,v_2,v_3,v_3\}|

    Same for S_4: |S_4| = 3 < 4

    For S_3 - what happens when you take the sum of its elements?

    S_1 is similar to S_3.

    For S_2, follow arbolis' advice.
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  6. #6
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    Quote Originally Posted by 450081592 View Post
    Suppose that {v1, v2, v3, v4} is a basis for a subspace V of  R^n. Which, if any, of the following subsets are bases for V?
    1) S1 = {v1 + v2, v2 + v3, v3 + v4, v4 + v1}
    2) S2 = {v1, v1 + v2, v1 + v2 + v3, v1 + v2 + v3 + v4}
    3) S3 = {v1 - v2, v2 - v3, v3 - v4, v4 - v1}
    4) S4 = {v1 + v2, v2+ v3, v3 + v4}
    5) S5 = {v1, v1 + v2, v2 + v3, v3 + v4, v1 + v2 + v3 + v4}
    Justify your answers?

    Can someone just show me how to do one of them, then I wil be able to do rest of them?
    I know a basis needs to be a span of V and linearly independent, I am not sure how to prove it is a span of V
    You do not need to show that they span V. Since V has a basis consisting of four vectors, it is of dimension 4 and so any set of 4 independent vectors in V is a basis for V.

    S3 has only three members and so cannot be a basis for V.

    S5 has five members and so cannot be a basis for V.

    The others have four vectors and so you only need to show that they are independent.
    To show that Si= {u1, u2, u3, u4} is independent, look at au1+ bu2+ cu3+ du4= 0. Rewrite them as a linear combination of v1, v2, v3, v4 and use the fact that v1, v2, v3, v4 are independent.

    For example, in S2, that would be a(v1)+ b(v1 + v2)+ c(v1 + v2 + v3)+ d(v1 + v2 + v3 + v4)= 0. That reduces to (a+ b+ c+ d)v1+ (b+ c+ d)v2+ (c+ d)v3+ dv4= 0. Since v1, v2, v3, and v4 are independent, we must have a+ b+ c+ d= 0, b+ c+ d= 0, c+ d= 0, and d= 0. Putting d= 0 into the third equation, c+ 0= 0 so c= 0. Putting d= 0 and c= 0 into the second equation, b+ 0+ 0= 0 so b= 0. Putting d= 0, c= 0, and b= 0 into the first equation, a+ 0+ 0+ 0= 0 so a= 0. We must have all coefficients zero so this set of four vectors is independent and forms a basis.
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  7. #7
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    Thanks a lot for the examples, such a great help !!!!!
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