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++:

标记,目前没有完成这道题,想出证明后会进行完善。

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.

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两个子空间。

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.