## 设U是V的子空间，求U + U

Question:

Suppose that $$U$$ is a subspace of $$V$$. What is $$U + U$$?

Solution:

$$U + U = U$$.

Proof:

Take $$u, v \in U$$. Then every element in $$U + U$$ can be written as $$u + v$$. Since $$U$$ is a subspace of $$V$$, U is closed under addition. Therefor $$u + v \in U$$. Hence $$U + U \subset U$$.

If we take $$w = u + 0 \in U + U$$, we can get every element in $$U$$ ($$w \in U$$). Therefor $$U \subset U + U$$.

Thus $$U = U + U$$.

## 证明V的三个子空间的并是V的子空间，当且仅当其中一个子空间包含另外两个子空间

Foreword++:

Foreword:

I strongly recommend you to read the previous blog which discussed a similar problem. You can go to that blog by clicking link: 证明V的两个子空间的并是V的子空间当且仅当其中一个子空间包含另一个子空间. I will skip the same part of this proof which is discussed in that proof.

Question:

Prove that the union of three subspaces of $$V$$ is a subspace of $$V$$ if and only if one of the subspaces of $$V$$ contains the others.

Solution:

Let $$A$$, $$B$$, $$C$$ be the three subspaces of V.

Part 1:

Assume $$A \cup B \cup C = A$$ or $$B$$ or $$C$$.

Clearly $$A \cup B \cup C$$ is one of these three vector spaces of $$A$$, $$B$$, $$C$$. Therefor $$A \cup B \cup C$$ is the subspaces of $$V$$.

Part 2:

Assume $$A \cup B \cup C \neq A$$ or $$B$$ or $$C$$.

Firstly, consider the condition:

$$A \cup B \cup C = A \cup B$$ or $$B \cup C$$ or $$A \cup C$$.

We have discussed this situation in the previous blog, you can read it by clicking this link: 证明V的两个子空间的并是V的子空间当且仅当其中一个子空间包含另一个子空间.

Secondly, consider the situation that neither of these three subspaces contained in the others.