What defines a linear mapping between vector spaces?

A linear mapping between vector spaces is defined by the properties of additivity and scalar multiplication. Specifically, a function ( T: V \rightarrow W ) (where ( V ) and ( W ) are vector spaces) is a linear mapping if for any vectors ( \mathbf{u}, \mathbf{v} \in V ) and any scalar ( c ), it satisfies two conditions:

1. Additivity: ( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) )
2. Homogeneity (Scalar Multiplication): ( T(c\mathbf{u}) = cT(\mathbf{u}) )

These properties ensure that the structure of the vector spaces is preserved under the mapping. This means that not only can you add vectors within the domain and map them appropriately, but you can also stretch or shrink vectors by a scalar and still maintain the mapping's integrity.

Other options do not capture the essence of what constitutes a linear mapping. For instance, while a function that only adds vectors might partially fulfill the criteria, it lacks the essential aspect of scalar multiplication, which is crucial for linearity. A function that always returns