Question

Vector Space Problem...

Answer 1

Let (V, +, ⋆) be a real vector space with the addition operation denoted
by + and the scalar multiplication operation denoted by ⋆. Let v 0 ∈ V be
fixed. We define a new addition operation ⊕ on V by x ⊕ y = x + y + v 0 ,
and a new scalar multiplication operation ⊛ by α ⊛ x = α ⋆ x + (α − 1) ⋆ v 0 .
Show that (V, ⊕, ⊛) defines a real vector space.

Answer 2

I think if you define two new operations which are different from the privous(+ and *), you need to verify six more laws to confirm whether (V, ⊕, ⊛) is the vector space.
According to Linear Algebra Done Right (the '+' and the '*' should be replaced by '⊕' and'⊛' )