Timeline for Interesting examples of vacuous / void entities
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 1 at 13:32 | history | edited | user3840170 | CC BY-SA 4.0 |
rescued dead link
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| May 13, 2020 at 14:31 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| May 5, 2011 at 11:38 | comment | added | Ramsay | There is a neat article by de Boor An empty exercise (doi), where he argues for the inclusion of $0\times n$ and $m \times 0$ matrices in Matlab. He discusses definitions of span and determinant etc. I found this from a link on the Wikipedia page for determinant. | |
| Nov 14, 2010 at 23:21 | history | edited | Richard Stanley | CC BY-SA 2.5 |
added 124 characters in body
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| Nov 14, 2010 at 22:35 | comment | added | Qiaochu Yuan | If you don't allow the empty set to span {0}, how on Earth are you going to define dim {0}? ({0} is not a basis for {0}, since 0 is a linear combination of elements of {0} in two ways.) | |
| Nov 14, 2010 at 11:50 | comment | added | darij grinberg | Is there a way to upvote the EC1 link without upvoting the whole posting? | |
| Nov 14, 2010 at 10:35 | comment | added | Laurent Moret-Bailly | Any reasonable definition of a span implies that the span of the empty set in a vector space is the zero subspace. No "convention" here: (1) It is the smallest vector subspace containing $\emptyset$. (2) It is the set of (finite) linear combinations of elements of $\emptyset$: the only such thing is the empty sum, which is 0. | |
| Nov 14, 2010 at 6:29 | comment | added | JBorger | I'm afraid I have to disagree with this answer. All sorts of things would break otherwise. For instance, for any set $S$, we want the $K$-vector space $K^S$ of functions $S\to K$ to have a the standard basis $\{e_s | s\in S\}$, where $e_s$ is the delta-function at $s$. When $S$ is empty, this just says that the empty set (which has zero elements) is a basis for the zero vector space (which has dimension zero). It would be a big nuisance to have say that the empty subset does not have a span. Then you'd have to add exceptions to lots of theorems that would otherwise be true without restriction. | |
| Nov 14, 2010 at 4:22 | comment | added | Tom Goodwillie | Example 3.10.3 gives a formula for the number of spanning sets in an n-dimensional vector space over a finite field when n>0. The fact that the formula gives 1 rather than 2 when n=0 could be taken as evidence that the empty set does not span the 0-dimensional space, but I would much prefer to say that Example 3.10.3 gives a formula for the number of nonempty spanning sets (for all n). | |
| Nov 14, 2010 at 3:18 | history | answered | Richard Stanley | CC BY-SA 2.5 |