Which axiom states that multiplication distributes over addition?

The axiom that states that multiplication distributes over addition is correctly identified as Axiom 7 (A7). This axiom articulates the distributive property, which expresses how multiplication interacts with addition in linear algebraic structures.

To clarify, the distributive property can be illustrated by the equation ( a(b + c) = ab + ac ), which shows that if a number ( a ) is multiplied by the sum of ( b ) and ( c ), it yields the same result as multiplying ( a ) by each term individually and then adding the results. This property is fundamental in both algebra and linear algebra because it allows for the simplification of expressions and is essential in solving equations and working with vectors and matrices.

The identification of Axiom 6, Axiom 5, or Axiom 8 does not accurately describe the distributive property, instead covering different properties within algebraic structures such as closure, associativity, or commutativity, which do not directly relate to the interaction of multiplication and addition. Understanding the specific role of Axiom 7 in the context of algebraic operations is crucial for mastering linear algebra concepts.