Yes, the set of all mxn matrices, denoted as Mmxn, is a vector space under matrix addition and scalar multiplication. To prove this, we need to show that the eight vector space axioms hold for Mmxn.
1. Closure under addition: For any two matrices A and B in Mmxn, their sum A + B is also a matrix in Mmxn.
2. Associativity of addition: For any matrices A, B, and C in Mmxn, (A + B) + C = A + (B + C).
3. Identity element of addition: There exists an identity matrix, denoted as 0, such that for any matrix A in Mmxn, A + 0 = A.
4. Inverse elements of addition: For any matrix A in Mmxn, there exists a matrix -A in Mmxn such that A + (-A) = 0.
5. Closure under scalar multiplication: For any scalar c and matrix A in Mmxn, their product cA is also a matrix in Mmxn.
6. Distributivity of scalar multiplication with respect to matrix addition: For any scalars c and d and matrix A in Mmxn, (c + d)A = cA + dA.
7. Distributivity of scalar multiplication with respect to scalar addition: For any scalar c and matrices A and B in Mmxn, c(A + B) = cA + cB.
8. Associativity of scalar multiplication: For any scalars c and d and matrix A in Mmxn, (cd)A = c(dA).
Since all of these eight axioms hold for Mmxn, we can conclude that Mmxn is indeed a vector space under matrix addition and scalar multiplication.
Let = the set of all mxn matrices. Is Mmxn a vector space under matrix addition and scalar multiplication?
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