【归档】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

Note: 旧的wordpress博客弃用，于是将以前的笔记搬运回来。

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Question:
Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces of $V$ is contained in the other.
证明V的两个子空间的并是B的子空间当且仅当其中一个子空间包含领一个子空间。
Solution:
Assume two set A, B are subspaces of $V$.
Part 1:
Assume $A\cup B=A$ or $B$.
Clearly $A\cup B=A$ or $B\in V$.
Therefor if one subspace of $V$ is contained in the other, the union of two subspaces is a subspace of $V$