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

[peripatein]

Hi,

Homework Statement

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

Homework Equations

The Attempt at a Solution

I proved that a group of vectors {u₁,...,u_k} in V is linearly independent if and only if {[u₁]_B,...[u_k]_B} is linearly independent. Following the principles of logic, that would be the same as proving dependence, right?

 

[LCKurtz]

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

 

[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?

 

[LCKurtz]

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?

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.