In checking if a transformation is linear, what should be verified first?

To determine if a transformation is linear, the first verification involves checking if the transformation correctly maps the zero vector. This is a fundamental property of linear transformations, as one of the defining characteristics is that a linear transformation ( T ) must satisfy two main conditions for any vectors ( u ) and ( v ), and any scalar ( c ):

1. Additivity: ( T(u + v) = T(u) + T(v) )
2. Homogeneity: ( T(cu) = cT(u) )

For both of these properties to hold true, the transformation must map the zero vector of the domain to the zero vector of the codomain. Specifically, this means ( T(0) ) must equal ( 0 ). If the transformation fails this initial test, it cannot be linear, regardless of whether it meets the other criteria.

Verifying the other aspects, such as whether the transformation is invertible, alters vector magnitudes, or maps all unit vectors, are important but can only be assessed after confirming that the transformation correctly handles the zero vector. Therefore, ensuring the proper mapping of the zero vector is the necessary first step in the process of testing for linearity.