Math Problem Solver

In the following, a set V , a field F, which is either R or C, and the operations of addition

and scalar multiplication are given. Check whether V is a vector space over F with these

operations.

(a). With addition and scalar multiplication as in R

2

, let V =

(a, b) ∈ R

2

: 2a + 3b = 0

and

F = R.

(b). Let V = R

2 and F = R. with addition as in R

2

, and scalar multiplication as given by the

: for (a, b) ∈ V, α ∈ R, α(a, b) := (a, 0).

(c). Let V = {x ∈ R : x > 0}, F = R, and for x, y ∈ V, α ∈ R, define the operations as follows:

x + y := xy, αx := x

α. The known operations on the right define the operations on the left.