What characterizes a linear transformation?

A linear transformation can be characterized by the property that it output a linear combination of its inputs. This means that if you have two vectors and you apply the transformation, the result can be expressed as a combination of the transformations of those vectors multiplied by scalars. This aligns with the formal definition of a linear transformation, which states that if T is a transformation from one vector space to another, then for any vectors u and v in the vector space, and any scalars a and b, the transformation must satisfy:

T(au + bv) = aT(u) + bT(v).

This property shows how linear transformations preserve the structure of vector addition and scalar multiplication, which is essential to their definition.

The other characteristics mentioned, such as preserving angles or always mapping vectors to zero, do not apply universally to all linear transformations. Preserving angles, for instance, describes orthogonal transformations specifically, while the transformation that maps all vectors to zero is a specific case (the zero transformation) rather than a defining trait of linear transformations in general. Similarly, stating that the transformation always increases the magnitude is inaccurate, as linear transformations can also decrease or maintain the magnitude depending on the input and the transformation applied.