## 证明或给出反例：如果U1, U2, W是V的子空间，使得U1 + W = U2 + W，则U1 = U2.

Question:

Prove or give a counterexample if $$U_1, U_2, W$$ are esubspaces of $$V$$ such that $$U_1 + W = U_2 + W$$, then $$U_1 = U_2$$.

Solution:

Let $$V = \mathbb{Z}^2$$, $$U_1 = 4\mathbb{Z}^2$$ and $$U_2 = W = 2\mathbb{Z}^2$$.

Clearly $$U_1 + W = U_2 + W = 2\mathbb{Z}^2$$, but $$U_1 \neq U_2$$.

For another example, we let $$U_1, U_2, W$$ be the spans of $$(0, 1), (1, 1), (1, 0)$$ respectively. Geometrically, $$U_1, U_2, W$$ are three mutually distinct lines. In this situation, we also have $$U_1 + W = U_2 + W = \mathbb{R}^2$$ but $$U_1 \neq U_2$$.

## V的子空间加法运算有单位元吗？哪些子空间有加法逆元？

Question:

Does the operation of addition on the subspaces of $$V$$ have an additive identity? Which subspaces have additive inverses?

Solution:

Obviously the subspace $$\{0\}$$ is an additive identity for the operation of addition on the subspace of $$V$$. Every subspace of $$V$$ plus $$\{0\}$$ is the subspace itself.

Assume $$U$$, $$V$$ are two subspaces of $$V$$. If $$V$$ is the additive inverse of $$U$$, $$U + V = \{0\}$$. Since both of them should be contained in the subspace $$U + V$$, they have to satisfy that $$U = V = \{0\}$$.