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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?

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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.

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