Linearly Independent Set Proof

Date: 01/24/2001 at 19:37:24
From: sanaz golban
Subject: Prove set is linearly independent

I have not been able to figure this one out. 

Assume that in a vector space V, the vectors u and v are linearly 
independent. Prove that the set {2u-v, u+5v} is linearly independent.

I have a hard time with proofs as well, so if you have any tips or 
steps that will help me solve any proofs, please let me know. 

Thank you. 

Date: 01/24/2001 at 19:48:12
From: Doctor Schwa
Subject: Re: Prove set is linearly independent

To prove something is linearly independent, you want to prove that
if a * (first vector) + b * (second vector) = 0, then a and b are both
zero.

Similarly by definition of linearly independent, you know that
if xu + yv = 0, then x = y = 0.

So, suppose  a(2u - v) + b(u + 5v) = 0.

By distributing, you can find that some combination of u and v is 
equal to zero, so you know that the coefficients are zero; that is, 
2a + b = 0 and -a + 5b = 0 also, from which you can prove that 
a = b = 0.

What method did I use for this proof?  I saw some complicated words,
"linearly independent."  Not only that, but those words were used
twice, so I wrote down the definition, and wrote down what the
"if-then" part of the definition said, and then found that I was 
already almost done with the proof.

Most proofs in linear algebra work pretty quickly that way; look up
the definition of the words in the proof, figure out what the "if"
part and the "then" part are (that is, what are the givens? what
is the goal?) and you're usually at least halfway done with
the proof.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/