To determine whether the given transformations are linear, we need to check if they satisfy two key properties of linear transformations:

1. Additivity: For vectors $\mathbf{u}$ and $\mathbf{v}$ in the vector space, the transformation must satisfy $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$.
2. Homogeneity (or scalar multiplication): For any vector $\mathbf{v}$ in the vector space and any scalar $c$, the transformation must satisfy $T(c\mathbf{v}) = cT(\mathbf{v})$.

Let's go through each transformation to check whether these conditions are satisfied.

i) $T: V_2 \rightarrow V_2$ defined by $T(4, 2) = (4 + 224)$

The transformation here seems to be unclear or not fully specified. However, I assume it's trying to define a transformation for vectors in $V_2$. Let's re-interpret the transformation as:

• If $T(4, 2) = (4 + 224)$, that would imply some operation on the vector $(4, 2)$ to produce the result (