Tv : R -> V is a linear transformation for each vector v in V. This is shown by verifying that Tv satisfies the properties of closure under addition and scalar multiplication. Additionally, every linear transformation T:R->V can be uniquely expressed as Tv for some vector v, as T(x) must equal x multiplied by the vector T(1).

1. Let V be a vector space. If v V define Tv : R -> V by Tv(x) = xv for all x in R. Show that Tv : R
-> V is a linear transformation for each v in V and show that every linear transformation T:R--
>V arises for a uniquely determined vector v in V.
Solution
To see that this is a linear transformation:
Closure under addition:
if x and y are two values in R:
T(x + y) = (x + y)v = xv + yv = T(x) + T(y)
So T is closed under addition.
Closure under multiplication:
if x is in R and k is a scalar
T(kx) = (kx)v = k(xv) = kT(x)
So T is closed under scalar multiplication.
Now for the second part: If T is a linear transformation, then T(1)=w for some vector w. Since T
is linear, for any x in R, T(x)=T(x * 1) = x T(1) = xw.
So T is uniquely defined by where it sends x.