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Math Help - Vectors and linear independence. help please.

  1. #1
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    Vectors and linear independence. help please.

    (a) If vectors u, v and w are linearly independent, will (u+v), (v+w) and (u+w) also be linearly independent? Justify your answer.

    (b)If vectors u, v and w are linearly independent, will (u-v), (v-w) and (u-w) also be linearly independent? Justify your answer.

    The answer in the back of the book is (a) yes. (b) no

    Any help on why this is?

    Cheers
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  2. #2
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    Quote Originally Posted by surfer101 View Post
    (a) If vectors u, v and w are linearly independent, will (u+v), (v+w) and (u+w) also be linearly independent? Justify your answer.

    (b)If vectors u, v and w are linearly independent, will (u-v), (v-w) and (u-w) also be linearly independent? Justify your answer.

    The answer in the back of the book is (a) yes. (b) no

    Any help on why this is?

    Cheers
    First recall what it means if vectors u, v and w are linearly independent. Now consider:

    (a) \alpha (u + v) + \beta (v + w) + \gamma (u + w) = (\alpha u + \beta v + \beta w) + (\gamma u + \alpha v + \gamma w) ....

    (b) \alpha (u - v) + \beta (v - w) + \gamma (u - w) = (\alpha u + \beta v + \beta w) - (\gamma u + \alpha v + \gamma w) ....

    where \alpha, \beta and \gamma are scalars.
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