When evaluating the linearity of a transformation after confirming it maps the zero vector, what is the next step?

To determine if a transformation is linear, confirming that it maps the zero vector is just the first step. The next crucial step is to test linear combinations of other vectors. This involves checking whether the transformation satisfies the properties of linearity:

1. Additivity: This means that for any two vectors u and v, the transformation T must satisfy T(u + v) = T(u) + T(v).
2. Homogeneity: This property indicates that for any scalar c and vector v, the transformation must hold that T(cv) = cT(v).

By evaluating these properties for arbitrary vectors, you establish that the transformation maintains the structure of vector addition and scalar multiplication, which are essential characteristics of linear transformations. Examining specific vectors and their combinations provides concrete evidence of the transformation's behavior with respect to linearity. This process confirms that the transformation can indeed be classified as linear if it adheres to both additivity and homogeneity.

The other options, while they touch on important concepts, do not directly pertain to confirming linearity following the check on the zero vector.