Introduction:
We are given a vector space V consisting of twice differentiable functions on R that satisfy the equation f' - 2f' + f = 0. We are also given a linear transformation T from V to R2 defined as T(f) = (f'(0), f(0)). We need to determine the nature of the linear transformation T.

Definition of Linear Transformation:
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Proving T is a Linear Transformation:
To show that T is a linear transformation, we need to prove two properties:

1. T(u + v) = T(u) + T(v) for all u, v in V (Preservation of vector addition)
2. T(cu) = cT(u) for all u in V and c in R (Preservation of scalar multiplication)

Preservation of Vector Addition:
Let f, g be two functions in V. We need to show that T(f + g) = T(f) + T(g).

T(f + g) = ((f + g)'(0), (f + g)(0)) [Using the definition of T]
= (f'(0) + g'(0), f(0) + g(0)) [By the linearity of differentiation]

T(f) + T(g) = (f'(0), f(0)) + (g'(0), g(0)) [Using the definition of T]
= (f'(0) + g'(0), f(0) + g(0)) [By vector addition]

Since T(f + g) = T(f) + T(g), the preservation of vector addition property holds.

Preservation of Scalar Multiplication:
Let f be a function in V and c be a scalar. We need to show that T(cf) = cT(f).

T(cf) = ((cf)'(0), (cf)(0)) [Using the definition of T]
= (cf'(0), cf(0)) [By the linearity of differentiation]

cT(f) = c(f'(0), f(0)) [Using the definition of T]
= (cf'(0), cf(0)) [By scalar multiplication]

Since T(cf) = cT(f), the preservation of scalar multiplication property holds.

Conclusion:
We have shown that T satisfies both properties of a linear transformation. Therefore, we can conclude that T is a linear transformation from V to R2.