What does a transformation matrix encode?

A transformation matrix encodes the rules for transforming a vector space through linear transformations. In linear algebra, a transformation matrix is a specific type of matrix that represents a linear transformation of vectors in a given vector space. When you multiply a transformation matrix by a vector, you apply the transformation to that vector, which could involve operations such as scaling, rotating, or translating the vector.

This concept is fundamental because linear transformations have crucial properties; they preserve vector addition and scalar multiplication. The transformation matrix effectively encodes how each vector in the space should be manipulated according to the defined linear transformation. Thus, the transformation matrix serves as a bridge between the abstract definition of a linear transformation and its concrete implementation in terms of matrix operations.

The other choices do not accurately describe the role of a transformation matrix. A transformation matrix is not merely a function for scalar multiplication, nor does it represent the process of adding matrices. Furthermore, it does not indicate a non-linear representation of vectors; instead, it is specifically tied to linear operations within a vector space.