Is M2 a Vector Space with Modified Scalar Multiplication?

[blazelian]

Is this a vector space?

Let M2 denote the set of all matrices of 2 x 2. Determine if M2 is a vector space when considered with the standard addition of vectors, but with scalar multiplication given by
α*(a b) = (αa b)
(c d) (c αd)
In case M2 fails to be a vector space with these definitions, list at least one axiom that fails to hold. justify you answer.

How do you solve this?

 

[pongo38]

Start with a good formal definition of a vector space.

 

[blazelian]

well a vector space is something tht looks like R^n

 

[Mark44]

well a vector space is something tht looks like R^n

That's not a formal definition. Your text should have a definition of a vector space, including about 10 axioms.

 

[DeeNos]

Yes, M2 is a vector space with the given definitions of addition and scalar multiplication. To prove this, we need to check if all the axioms of a vector space hold for M2.

1. Closure under addition: For any two matrices A = (a b) and B = (c d) in M2, A+B = (a+c b+d) is also a matrix in M2. Therefore, M2 is closed under addition.

2. Associativity of addition: (A+B)+C = (a+c b+d)+(e+f g+h) = (a+e+c f) + (b+g+d h) = A+(B+C). Therefore, addition is associative in M2.

3. Commutativity of addition: A+B = (a+c b+d) = (c+a d+b) = B+A. Therefore, addition is commutative in M2.

4. Existence of additive identity: The zero matrix O = (0 0) is the additive identity in M2 since A+O = (a+0 b+0) = A for any A = (a b) in M2.

5. Existence of additive inverse: For any A = (a b) in M2, the additive inverse is -A = (-a -b) since A+(-A) = (a+(-a) b+(-b)) = (0 0) = O.

6. Closure under scalar multiplication: For any scalar α and matrix A = (a b) in M2, α*A = (αa b) is also a matrix in M2. Therefore, M2 is closed under scalar multiplication.

7. Distributivity of scalar multiplication over vector addition: α*(A+B) = α*((a+c b+d)) = (αa+αc αb+αd) = (αa αb)+(αc αd) = α*A+α*B.

8. Distributivity of scalar multiplication over scalar addition: (α+β)*A = ((α+β)a (α+β)b) = (αa+βa αb+βb) = (αa αb)+(βa βb) = α*A+β*A.

9. Associativity of scalar multiplication: (αβ)*A = ((αβ)a (αβ)b) = (α(βa) β(β