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I need to find two subsets of $\mathbb{R}^3$ whose spans equal each other but their intersection is the empty set. I was thinking $v_1=\{(1,0,0)\}$ and $v_2=\{(0,1,0)\}$ but I'm not sure..would this work?

Cure
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Aplusmath
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    The reason your example won't work is because these vectors are orthogonal and hence have mutually orthogonal spans. – Squirtle May 05 '14 at 02:05
  • please consider selecting an answer as your favorite by ticking the check mark at the top left of the answers. – Squirtle May 06 '14 at 13:35

3 Answers3

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Christopher's answer is really good but I enjoy giving off the wall answers:

Let the first set be the empty set and the second set be $\{0\}$, their intersection is the empty set and their span is the same because Span of an empty set is the zero vector

Community
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Squirtle
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  • I feel that's a somewhat confusing definition, since the 0 vector can itself be considered a subspace. – Christopher Liu May 05 '14 at 02:10
  • Read the top voted answer on that page, I agree that this is an unusual answer but.... but applying the definitions I'm completely correct. – Squirtle May 05 '14 at 02:11
  • @Christopher, (0) is a vector subspace, while {0} is the vector which generates the former space. – Squirtle May 05 '14 at 02:20
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Hint: You just need two non-zero vectors where one is a scalar multiple of the other. Can you see why?

In your attempt, $v_1$ and $v_2$ do not have the same span.

Christopher Liu
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Does it looks like the span of $v_1$ and $v_2$ is the same?. This certainly wouldn't work, $v_1$ generates the set of $\{(a,0,0):a\in\mathbb{R}\}$ and $v_2$ generates $\{(0,b,0):b\in\mathbb{R}\}$ that only have in common the null vector.

I'll give you and intuitive hint: How many ways do you have to span a line or a plane?, how can you set the vectors to span them?. Consider for example a line that goes along the x-axis, would it make a difference if you define de subspace using a vector that points in to the negative direction instead of the positive?.

Cure
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  • Kind of a side question: if two subsets of a linear space have the same span, does it mean they are always linear dependent? – Kemeia May 28 '19 at 08:41