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 \(U = V = \{0\}\).