# {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.