Showing subspace is a vector space. Why is this step necessary?

Question

I am reading this text:

Why do we need to show the transpose of the sum is equal to the sum? Isn't it enough to just show that W, the set of of all 2x2 symmetric matrices, is closed under addition and multiplication? I realize symmetric means that the transpose is equal to the original matrix, but we don't need to show this right to show closure under addition right?

Answer 1

The author of that text needed to show that $A_1+A_2$ is symmetric, assuming that $A_1$ and $A_2$ are symmetric. So, since $A$ being symmetric means that $A^T=A$, he or she proved that $(A_1+A_2)^T=A_1+A_2$. That's all. The same thing applies to the product by a scalar.

Answer 2

That is exactly how you prove the closure under addition and scalar multiplication.j

You show that if you add two symmetric matrices, the sum is symmetric, that is $$(A+B)^T= A+B$$ and $$(\lambda A)^T=\lambda A $$ and that is what the author is doing.