What is a linear transformation?

A linear transformation is defined as a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you take two vectors and apply the linear transformation, the result will be the same as if you first added the vectors and then applied the transformation. Additionally, if you scale a vector by a constant and then apply the transformation, the outcome will be the same as if you applied the transformation first and then scaled the result.

This property is essential because it captures the fundamental characteristics of linear systems and allows for consistent behavior in mathematical modeling and analysis. For instance, in two-dimensional space, linear transformations encompass operations such as rotations, reflections, scalings, and shears, but are not limited to just those operations; they are applicable in any dimension.

In contrast, other options suggest limitations or incorrect definitions. One option incorrectly limits the transformation to rotations specifically, another limits it to only two-dimensional spaces, and the last one defines it as solely concerned with changing vector lengths, which does not accurately represent the broader scope of what linear transformations can do. Thus, the comprehensive nature of option A accurately reflects the foundational concept of linear transformations in linear algebra.