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Let u and v be non-zero vectors in V. Prove or disprove the following claim. u and v are linearly dependent ⟹ (u+v) and (u−v) are linearly dependent.
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.
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.
