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

1. Dec 10, 2012

peripatein

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. Dec 10, 2012

LCKurtz

Yes, and it is an easy one-liner to prove it instead of just declare it.

3. Dec 10, 2012

peripatein

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?

4. Dec 10, 2012

LCKurtz

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.