This is what I eventually came up with and I got confirmation from a member of another math forum.
Let A = span\{\left[\begin{array}{c}1\\0\\0\end{array}\right] \left[\begin{array}{c}0\\0\\1\end{array}\right]\}.
Let B = span\{\left[\begin{array}{c}0\\1\\0\end{array}\right]...
The point (1, 1, 1) is not the same as the vector <1, 1, 1>. If the planes only meet at one particular point, then their intersection is just the zero vector. So the intersection of these 3 planes is still just the zero vector. I need the 3 planes to intersect in a line segment, not just one...
In this situation the subspaces A, B, and C do not satisfy the first requirement: A \cap B \cap C \neq \{\vec{0}\}. The only vector in common among the 3 planes is the zero vector. But the problem is asking for subspaces A, B, and C, that intersect in more than just the case of the zero vector.
Homework Statement
Find subspaces A, B, and C of \mathbb{R}^3 so that A \cap B \cap C \ne \{\vec{0}\} and (A + B) \cap C \ne A \cap C + B \cap C.
You can specify a subspace by the form A = span\{\vec{e}_1, \vec{e}_2\}.Homework Equations
A + B is the set of all vectors in \mathbb{R}^3 of the...
Ok, subspaces of \mathbb{R}^3 have the following properties: contain the zero vector, are closed under addition, and are closed under multiplication. Am I right?
So, say A, B, and C are subspaces of \mathbb{R}^3. Then, what would (A+B) \cap C mean?