Matrix Subspace question: Does B2x2 form a subspace of M2x2?

[OFFLINEX]

Homework Statement

The set of all matrices A2x2 forms a vector space under the normal operations of matrix + and Scalar multiplication. Does the set B2x2 of all symmetric matrices form a subspace of M2x2? Explain.

Homework Equations

AT = A
Closure property of addition - If w and v are objects in A, then w+v are contained within A
Closure property of scalar multiplication - If K is any real number scalar and v is any object in A, then kv is also in A

The Attempt at a Solution

To be a subspace it must pass two axioms

1. B2x2 = [a,b;b,d] [a,b;b,d] + [e,f;g,h] = [a+e,b+f;b+g,d+h] I thought that this failed because it was not symmetric, but does it matter if the answer isn't symmetric only that its contained within A2x2

2. Scalar multiplication passes

 

[Dick]

Um, if w is B it has the form [[w1,w2],[w2,w3]]. If v is in B it has the form [[v1,v2],[v2,v3]. Add them. I'm not sure what you are on about with addition.

 

[OFFLINEX]

Well my main question is, would B2x2 pass the closure property of addition because when you add another thing to it that's not symmetric it, it looses its symmetry. Would that matter.

So does it pass that axiom or not is what I'm asking.

 

[Mark44]

Well my main question is, would B2x2 pass the closure property of addition because when you add another thing to it that's not symmetric it, it looses its symmetry. Would that matter.

It doesn't matter, since the things that are in B2x2 are symmetric matrices. One of the things you're checking to show closure under "vector" addition is whether adding two symmetric 2 x 2 matrices gives you a symmetric 2x2 matrix.

So does it pass that axiom or not is what I'm asking.