Final answer:
The set M(m x n; R) is a vector space because it satisfies the axioms of commutativity, associativity of addition, and distributivity of scalar multiplication, along with having a zero matrix and additive inverses.
Step-by-step explanation:
The question asks us to show that the set M(m \times n; \mathbb{R}), which consists of all m \times n matrices with entries from the set of real numbers (\mathbb{R}), forms a vector space under matrix addition and scalar multiplication. A vector space over a field F (like \mathbb{R}) must satisfy several axioms, such as commutativity and distributivity of vector addition, and compatibility of scalar multiplication with field multiplication.
For matrices, commutativity of addition is given by A + B = B + A, where A and B are matrices. Associativity of vector addition for matrices is shown by (A + B) + C = A + (B + C), and the distributive property of scalar multiplication over vector addition is demonstrated by c(A + B) = cA + cB, where 'c' is a scalar from \mathbb{R} and A and B are matrices. These properties, along with the existence of a zero matrix (acting as the additive identity) and additive inverses (the negative of a given matrix), ensure that M(m \times n; \mathbb{R}) qualifies as a vector space.