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{u1, ,uk} is linearly dependent iff {[u1]_B, [uk]B} is.

  1. Dec 10, 2012 #1
    Hi,
    1. The problem statement, all variables and given/known data

    Given a vector space V and its basis B = {v1, v2, ..., vn}, I was asked to prove that:
    a group of vectors {u1,...,uk} in V is linearly dependent if and only if {[u1]B,...[uk]B} is linearly dependent.

    2. Relevant equations



    3. The attempt at a solution

    I proved that a group of vectors {u1,...,uk} in V is linearly independent if and only if {[u1]B,...[uk]B} is linearly independent. Following the principles of logic, that would be the same as proving dependence, right?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 10, 2012 #2

    LCKurtz

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    Yes, and it is an easy one-liner to prove it instead of just declare it.
     
  4. Dec 10, 2012 #3
    So, in general, proving (A if and only if B) is equivalent to, and hence may be proven by, proving (-A if and only if -B), correct?
     
  5. Dec 10, 2012 #4

    LCKurtz

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    Yes. But declaring it doesn't make it so. If I were handing in a homework proof of A iff B and I instead handed in a proof that -A iff -B, I wouldn't just stop there even though you might consider the conclusion to be trivial from there.
     
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