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Math Help - Linearly Independents Sums and Differences

  1. #1
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    Linearly Independents Sums and Differences

    I am having trouble understanding the answers to the following two questions:

    1. If vector u, v, and w are linearly independent, will u+v, v+w, and u+w also be linearly independent?
    2. If vector u, v, and w are linearly independent, will u-v, v-w, and u-w also be linearly independent?


    I know the answer to the first question is yes, and I know the answer to the second question is no, but I don't know why. Can someone provide me with an explanation. I can't come up with any justifications.
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  2. #2
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    Quote Originally Posted by 323k13l View Post
    I am having trouble understanding the answers to the following two questions:

    1. If vector u, v, and w are linearly independent, will u+v, v+w, and u+w also be linearly independent?
    2. If vector u, v, and w are linearly independent, will u-v, v-w, and u-w also be linearly independent?


    I know the answer to the first question is yes, and I know the answer to the second question is no, but I don't know why. Can someone provide me with an explanation. I can't come up with any justifications.
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  3. #3
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    Hello,

    The definition of linearly independent vectors u, v & w is :

    If there are a,b,c three scalars such as au+bv+cw=0, u, v and w are l-i if and only if a=b=c=0

    What if a'(u+v)+b'(v+w)+c'(u+w)=0 ? Put the coefficients of u, v and w together. Are a', b' and c' necessarily null ?

    The same for a'(u-v)+b'(v-w)+c'(u-w)=0, is there a case when a', b' and c' are not null, but where the sum is null ?
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    Thank you to both of you. It's obvious now, and it makes sense.
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