if λ v=0,the neither λ=0 or v=0. Pls help me prove this

Question

 if λ v=0,the neither λ=0 or v=0.

Answer 1

The statement should say EITHER, not NEITHER.

 

Assuming λ is a real scalar and v is a vector in Rⁿ,

 

λv = (0,0,0,...,0)  [given]

 

λv = λ(v₁,v₂,...,v_n) = (λv₁,λv₂,...,λv_n) = (0,0,...,0) ⇒

 

λv_i = 0 for all i

 

Then either 

 

λ = 0

 

or

 

v_i = 0 for all i.

 

The latter ⇒ v = (0,0,....0) = 0

 

Conclusion:  Either λ = 0 or v = 0