Linear Transformation:

A linear transformation is a function that maps vectors from one vector space to another, while preserving the operations of addition and scalar multiplication. In other words, if T is a linear transformation, it satisfies the following properties:

1. T(u + v) = T(u) + T(v) for all vectors u and v in the domain of T.
2. T(cu) = cT(u) for all vectors u in the domain of T and scalar c.

Nullity of a Linear Transformation:

The nullity of a linear transformation T, denoted by nullity(T), is the dimension of the null space of T. The null space of T is the set of all vectors in the domain of T that are mapped to the zero vector in the codomain.

Given Linear Transformation:

Let's consider the given linear transformation T: R² → R³ defined as T(a,b) = (a b, a-b, b).

Determining the Null Space:

To find the null space of T, we need to find all vectors (a,b) in R² such that T(a,b) = (0 0 0).

T(a,b) = (a b, a-b, b) = (0 0 0)

By comparing the components, we have:
a = 0
b = 0
a - b = 0

From the first two equations, we can see that a = b = 0. Substituting these values in the third equation, we have:
0 - 0 = 0

Therefore, the only solution is (a,b) = (0,0).

Nullity of T:

The null space of T consists only of the zero vector (0,0). Since the null space only contains the zero vector, it is a trivial subspace of dimension 0.

Hence, the nullity of T is 0.

Summary:

In summary, the given linear transformation T: R² → R³ defined as T(a,b) = (a b, a-b, b) has a nullity of 0. This means that the null space of T only contains the zero vector, indicating a trivial subspace.