(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find subspaces [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] of [tex]\mathbb{R}^3[/tex] so that [tex]A \cap B \cap C \ne \{\vec{0}\}[/tex] and [tex](A + B) \cap C \ne A \cap C + B \cap C[/tex].

You can specify a subspace by the form [tex]A = span\{\vec{e}_1, \vec{e}_2\}[/tex].

2. Relevant equations

[tex]A + B[/tex] is the set of all vectors in [tex]\mathbb{R}^3[/tex] of the form [tex]\vec{a} + \vec{b}[/tex], where [tex]\vec{a}[/tex] is in [tex]A[/tex] and [tex]\vec{b}[/tex] is in [tex]B[/tex].

3. The attempt at a solution

I’ve been trying to guess. I’ve tried [tex]A[/tex] as a line, [tex]B[/tex] as a plane containing [tex]A[/tex], and [tex]C[/tex] as [tex]\mathbb{R}^3[/tex] itself. I’ve tried making all three subspaces planes that intersect in a line. I’ve tried making [tex]A[/tex] and [tex]B[/tex] planes that intersect in a line and C as [tex]\mathbb{R}^3[/tex] itself. I’ve tried making [tex]A[/tex] equal to [tex]B[/tex] and [tex]C[/tex] another plane intersecting them in a line.

Finding [tex]A \cap B \cap C \ne \{\vec{0}\}[/tex] is not my problem. I’m having trouble satisfying the second specification: [tex](A + B) \cap C \ne A \cap C + B \cap C[/tex].

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# Homework Help: Find the following subspaces

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