#### Snitch

##### Junior Member level 3

Is the following proof correct? Otherwise can someone answer the given question? Thanks in advance.

Here is my answer.

Since u,v are two linearly dependent vectors, au+bv=0 with a,b≠0

we can write, [(a+b)/2]⋅(u+v)+[(a−b)/2]⋅(u−v)=0

case 1 (|a|=|b|) :

if a=b then (a+b)/2≠0⟹(u+v) and (u−v) are linearly dependent.

if a=−b then (a−b)/2≠0⟹(u+v) and (u−v) are linearly dependent.

case 2 (|a|≠|b|) :

then (a+b)/2,(a−b)/2≠0⟹(u+v) and (u−v) are linearly dependent.

So, u and v are linearly dependent ⟹(u+v) and (u−v) are linearly dependent.