Which axiom expresses that multiplication operates without altering summation?

The correct answer is based on the understanding of how multiplication interacts with addition in the context of vector spaces and linear transformations. The axiom that describes multiplication operating without altering summation is Axiom 7, commonly known as the distributive property. This axiom states that for any scalars and vectors, the multiplication of a scalar with the sum of two vectors can be distributed over that sum. In mathematical terms, this can be expressed as:

c(u + v) = cu + cv

where c is a scalar and u and v are vectors. This property is fundamental in linear algebra because it ensures that when you multiply a sum of vectors by a scalar, it is equivalent to multiplying each vector by the scalar and then summing the results.

This concept is vital for maintaining the structure of vector spaces and ensures that operations involving vectors and scalars behave in a consistent manner according to the mathematical rules governing these operations. Therefore, Axiom 7 captures the essence of how multiplication and addition interact in this framework.