# Sum of subspaces confusion

I have been confused on this for a while. If we need to show that a certain vector space is the sum of two of its subspaces, I am confused on the proof methods.

Every time I post a question on this people tell me that I need to show the set equality by proving the subset both directions.

However I often see solutions to these problems done a different way. They show a generic vector in the vector space can be decomposed into two vectors in the subspaces.

For instance the following problem:

Let $$W_1$$ be the subspace of $$\mathcal{M}_{n \times n}$$ that consists of all $$n \times n$$ skew-symmetric matrices with entries from $$\mathbb{F}$$, and let $$W_2$$ be the subspace of $$\mathcal{M}_{n \times n}$$ consisting of all symmetric $$n \times n$$ matrices. Prove that $$\mathcal{M}_{n \times n}(\mathbb{F}) = W_1 \oplus W_2$$.

Proof: $$M = \frac{1}{2}(M + M^{t}) + \frac{1}{2}(M-M^{t}),$$ where $$\frac{1}{2}(M+M^{t}) \in W_2, \frac{1}{2}(M-M^t) \in W_1$$, and $$M \in \mathcal{M}_{n \times n}(\mathbb{F})$$

Thus $$\mathcal{M}_{n\times n}=W_1+W_2$$

Clearly this problem was not solved by showing $$\mathcal{M}_{n\times n}\subset W_1+W_2$$ and $$\mathcal{M}_{n\times n}\supset W_1+W_2$$

This makes me very confused. What are the conventions for proving a vector space is the sum of two subspaces? Also, when do we have to prove the subset in both directions vs. showing a generic vector in the vector space can be decomposed as the sum of two elements of the subspaces?

Since the left side is the entire space of $$n \times n$$ matrices we don't have to prove that RHS is contained in LHS. That is automatic here. That is why we start with $$M$$ in the left side and show that it belongs to RHS. But in general you have to prove that each side is contained in the other.