Let = the set of all mxn matrices. Is Mmxn a vector space under matrix addition and scalar multiplication?
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.
To determine whether the set of all mxn matrices, denoted as Mmxn, is a vector space under matrix addition and scalar multiplication, we need to check if it satisfies the ten axioms of a vector space.
1. Closure under matrix addition: For any two matrices A and B in Mmxn, the sum A + B is also a matrix in Mmxn.
2. Commutativity of matrix addition: For any two matrices A and B in Mmxn, A + B = B + A.
3. Associativity of matrix addition: For any three matrices A, B, and C in Mmxn, (A + B) + C = A + (B + C).
4. Existence of zero matrix: There exists a zero matrix, denoted by 0, such that for any matrix A in Mmxn, A + 0 = A.
5. Existence of additive inverse: For every matrix A in Mmxn, there exists a matrix -A in Mmxn such that A + (-A) = 0.
6. Closure under scalar multiplication: For any scalar c and any matrix A in Mmxn, the product cA is also a matrix in Mmxn.
7. Associativity of scalar multiplication: For any scalar c and any matrices A and B in Mmxn, (cA)B = c(AB).
8. Distributivity of scalar multiplication over matrix addition: For any scalars c and d and any matrix A in Mmxn, (c + d)A = cA + dA.
9. Distributivity of scalar multiplication over scalar addition: For any scalar c and any matrices A and B in Mmxn, c(A + B) = cA + cB.
10. Compatibility of scalar multiplication with field multiplication: For any scalars c and d and any matrix A in Mmxn, c(dA) = (cd)A.
By verifying each of these axioms, we can determine if Mmxn is a vector space under matrix addition and scalar multiplication.
Yes, the set of all mxn matrices, denoted by Mmxn, forms a vector space under matrix addition and scalar multiplication. Here are the step-by-step properties that need to be satisfied for a set to be a vector space:
1. Closure under addition: For any two matrices A and B in Mmxn, the sum A + B is also an mxn matrix.
2. Commutative property of addition: For any matrices A and B in Mmxn, A + B = B + A.
3. Associative property of addition: For any matrices A, B, and C in Mmxn, (A + B) + C = A + (B + C).
4. Identity element of addition: There is an mxn matrix called the zero matrix, denoted as 0, such that for any matrix A in Mmxn, A + 0 = A.
5. Additive inverse: For any matrix A in Mmxn, there is an mxn matrix called the additive inverse of A, denoted as -A, such that A + (-A) = 0.
6. Closure under scalar multiplication: For any scalar c and any matrix A in Mmxn, the product cA is also an mxn matrix.
7. Associative property of scalar multiplication: For any scalar c1 and c2 and any matrix A, c1(c2A) = (c1c2)A.
8. Distributive properties: For any scalars c1 and c2 and any matrix A in Mmxn, (c1 + c2)A = c1A + c2A and c1(A + B) = c1A + c1B.
9. Scalar multiplication identity: For any matrix A in Mmxn, 1A = A, where 1 is the multiplicative identity of scalars.
By verifying these nine properties, we can conclude that Mmxn is indeed a vector space under matrix addition and scalar multiplication.