Given a set S = {(a, b) | a, b ∈ ℝ} and have a sum definition as of vector space R2, and a multiplication scalar as k(a, b) = ((k.a) + 1, kb) whereas k is a scalar.
Given those circumstances is set S a vector space or not?
In the definition from my lecturer, k.u = k(a, b) has to be (k.a, k.b) thus, k.u ∈ S. That concludes S is not a vector space because it doesn't meet the axiom of scalar multiplication, note that k.a+1 != k.a
But, what confused me is k.a + 1 is also a real number(a, b ∈ ℝ). So in my opinion,
k(a, b) = ((k.a) + 1, k.b) is also ∈ S
Which one is true?
$k(l(a,b))=k(la+1,lb)=(kla+k+1, klb),$ while $(kl)(a,b)=(kla+1,klb)$.