verify The set of all 2 × 2 invertible matrices with the standard matrix addition and scalar multiplication is a vector space or not?
-1
down vote
The set of all invertible n×nn×n matrices is not a vector space with respect to the typical matrix addition and scalar multiplication operations and the typical matrix zero.
However,
The special orthogonal group (rotation matrices) is a vector space if you use matrix multiplication for the addition operator and the identity matrix as the zero matrix.
You also have to make scalar multiplication exponentials.
You can also change the zero matrix to be a matrix CC with:
A⊕B=ABC−1A⊕B=ABC−1
and
s⊗A=AsC−ss⊗A=AsC−s.
Then,
A⊕C=AA⊕C=A.
And invertible n×nn×n matrices are just rotation matrices with a scaling part and so are also vector spaces.
its very trickyy but easy
To verify if the set of all 2x2 invertible matrices forms a vector space, we need to check if it satisfies the axioms of a vector space.
1. Is the set closed under addition?
For any two invertible matrices A and B, if we add them together, A+B, the result is another 2x2 matrix. To show that the set is closed under addition, we need to demonstrate that the sum of any two invertible matrices is also invertible.
To prove this, we can use the fact that if a matrix is invertible, then its determinant is non-zero. Let's assume A and B are invertible matrices, their determinants are non-zero, and we want to show that A+B is also invertible. Suppose we have a matrix X such that (A+B)X = 0, where 0 is the zero matrix. We can multiply on the left by the inverse of (A+B), denoted by (A+B)^(-1), to get X = 0. Since X = 0 is the only solution, we can conclude that the inverse of (A+B) exists, thus making A+B invertible.
2. Is the set closed under scalar multiplication?
For any invertible matrix A and a scalar c, if we multiply A by c, the result is another 2x2 matrix. To show that the set is closed under scalar multiplication, we need to demonstrate that cA is also invertible.
Let A be an invertible matrix, and suppose there exists a non-zero scalar c such that cA is not invertible. This would imply that (cA)^(-1) does not exist, which means there exists a non-zero matrix X such that (cA)X = 0. However, by associativity, we can rewrite this equation as c(A(X)) = 0, which contradicts the fact that A is invertible since A(X) = 0 has only the trivial solution. Therefore, cA must also be invertible.
3. Is there an additive identity?
The additive identity is the matrix that, when added to any other matrix in the set, leaves the other matrix unchanged. In this case, the zero matrix (the matrix with all entries equal to zero) serves as the additive identity.
4. Does each matrix have an additive inverse?
For every invertible matrix A, there exists another invertible matrix -A (the negative of A) such that A+(-A) = 0. In this case, the inverse of A itself serves as its additive inverse.
5. Is scalar multiplication associative and distributive?
Scalar multiplication in this set is associative and follows the usual properties of matrix multiplication. It is also distributive over both matrix addition and scalar addition.
Since the set of all 2x2 invertible matrices satisfies all the required axioms of a vector space, we can conclude that it is indeed a vector space.
To verify if the set of all 2×2 invertible matrices, denoted as M, with standard matrix addition and scalar multiplication forms a vector space, we need to check if it satisfies the ten axioms of a vector space.
1. Closure under addition: Take any two invertible matrices A and B from M. The sum of A and B, denoted as A + B, is also a 2×2 invertible matrix. Therefore, closure under addition holds.
2. Commutativity of addition: Take any two invertible matrices A and B from M. A + B is equal to B + A since matrix addition is commutative. Therefore, commutativity of addition holds.
3. Associativity of addition: Take any three invertible matrices A, B, and C from M. (A + B) + C is equal to A + (B + C) since matrix addition is associative. Therefore, associativity of addition holds.
4. Identity element of addition: The identity element of addition, denoted as 0, is the 2×2 zero matrix. For any invertible matrix A from M, A + 0 = A since adding the zero matrix does not change A. Therefore, the identity element of addition holds.
5. Inverse elements of addition: For any invertible matrix A from M, the additive inverse of A, denoted as -A, exists in M since the inverse of an invertible matrix is also invertible. Therefore, all matrices in M have additive inverses, and the inverse elements of addition hold.
6. Closure under scalar multiplication: Take any invertible matrix A from M and a scalar c. The product of A and c, denoted as cA, is a 2×2 invertible matrix. Therefore, closure under scalar multiplication holds.
7. Distributivity of scalar multiplication with respect to matrix addition: Take any invertible matrices A and B from M, and a scalar c. c(A + B) is equal to cA + cB since scalar multiplication distributes over matrix addition. Therefore, distributivity of scalar multiplication with respect to matrix addition holds.
8. Distributivity of scalar multiplication with respect to scalar addition: Take any invertible matrix A from M, and scalars c and d. (c + d)A is equal to cA + dA since scalar multiplication distributes over scalar addition. Therefore, distributivity of scalar multiplication with respect to scalar addition holds.
9. Associativity of scalar multiplication: Take any invertible matrix A from M, and scalars c and d. (cd)A is equal to c(dA) since scalar multiplication is associative. Therefore, associativity of scalar multiplication holds.
10. Identity element of scalar multiplication: The identity element of scalar multiplication, denoted as 1, is the scalar value 1. For any invertible matrix A from M, 1A = A since multiplying by 1 does not change A. Therefore, the identity element of scalar multiplication holds.
Since the set of all 2×2 invertible matrices with standard matrix addition and scalar multiplication satisfies all ten axioms of a vector space, we can conclude that it is indeed a vector space.