Need help with linear independence proof

[dyanmcc]

Hi,

I don't know how to do the following proof:

If (v1, ...vn) are linearly independent in V, then so is the list (v1-v2, v2-v3, ...vn-1 -vn, vn).

I can do the proof if I replace 'linearly independent' with 'spans V' ...so what connection am I missing?

Thanks much!

 

[0rthodontist]

Prove it by contradiction. If the second set of vectors was not linearly independent, then you can write 0 as a linear combination of those vectors. Then simply expand out each vector to show that this implies v1...vn are also linearly dependent.

 

[dyanmcc]

Great thanks. Here's another one for you...Prove that if V is finite dimensional with dim V > 1, then the set of noninvertible operators on V is not a subspace of L(V)

 

[AKG]

What have you done for that second problem dyanmcc? Start by thinking about matrices.

 

[0rthodontist]

Prove it by contradiction. If the second set of vectors was not linearly independent, then you can write 0 as a linear combination of those vectors. Then simply expand out each vector to show that this implies v1...vn are also linearly dependent.

Also be sure to prove that there is still a nonzero coefficient when you expand the vectors out. (look at the FIRST nonzero coefficient before expansion)