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I need some info on bases but without the restriction of any vector having a unique representation, so, for example, if we have a vector space $V$ over a field $F$ with a, let's call it, a "semi-basis" $SB=\{e_1,\ldots,e_n\}$, that means that they are linearly independent of each other and every vector $v$ can be represented as linear combination of elements of the semibasis, so, for any $v\in V$, if $e_1,\ldots,e_n\in SB$,

$$v=a_1e_1+\cdots+a_ne_n$$

but it may be that

$$v=a_1e_1+\cdots+a_ne_n=b_1e_1+\cdots+b_ne_n$$

where $a_i\neq b_i$ for some $1\leq i\leq n$. Is there a name for a basis like this? If in vector spaces this condition is impossible and it needs a module or something like that, I consider an answer for them as valid. Thanks.

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    $\begingroup$ A spanning set of vectors. $\endgroup$ Nov 12 '18 at 15:01
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    $\begingroup$ If elements in the set are linearly independent, and every vector can be represented as a linear combination of elements in the set, then one can show that the representation must be unique. The official name for such set is "basis". $\endgroup$
    – user587192
    Nov 12 '18 at 15:05
  • $\begingroup$ Non-uniqueness of representation in this spanning set implies linear dependence. Note that, if $v = a_1e_1 + \ldots + a_n e_n$, then $e_1\left(a_1 - b_1\right) + \ldots + \left(a_n - b_n\right)e_n = 0$, and this means that the vectors $\{e_1,\ldots,e_n\}$ are linearly dependent. This is very important. $\endgroup$ Nov 12 '18 at 15:05
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Yes, it has a name. It is a “spanning set”.

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  • $\begingroup$ Would it be a linearly independent spanning set? $\endgroup$
    – Garmekain
    Nov 12 '18 at 15:02
  • $\begingroup$ A linearly independent spanning set is a basis. $\endgroup$ Nov 12 '18 at 15:03
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    $\begingroup$ @Garmekain Demanding that the spanning set is linearly independent is equivalent to demanding that every vector has a unique representation. $\endgroup$
    – 5xum
    Nov 12 '18 at 15:04
  • $\begingroup$ @5xum Is it just in vector spaces? What about modules or semimodules? $\endgroup$
    – Garmekain
    Nov 12 '18 at 15:08
  • $\begingroup$ @Garmekain Isn't your question about vector spaces? $\endgroup$ Nov 12 '18 at 15:08
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If the vectors are linearly independent and they span $V$ the representation for any vector $v$ is unique that is

$$v=a_1e_1+\cdots+a_ne_n=b_1e_1+\cdots+b_ne_n \iff a_i=b_i$$

and it is by definition a basis, indeed

$$a_1e_1+\cdots+a_ne_n=b_1e_1+\cdots+b_ne_n $$

$$\iff (a_1-b_1)e_1+\cdots+(a_n-b_n)e_n=0 \iff a_i=b_i$$

Otherwise, if the span $V$ and are not linearly independent, we have infinitely many representation for any vector $v$ (assuming the field F infinite) and we define it as a spanning set.

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  • $\begingroup$ "[...] we have infinitely many representation[s]" : assuming the field $F$ is infinite. $\endgroup$
    – Arnaud D.
    Nov 12 '18 at 15:23
  • $\begingroup$ @ArnaudD. Yes you are right, I'm referring to that case! In general we can say that we do not have an unique representation. $\endgroup$
    – user
    Nov 12 '18 at 15:24
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... if we have a vector space $V$ over a field $F$ with a, let's call it, a "semi-basis" $SB=\{e_1,\ldots,e_n\}$, that means that they are linearly independent of each other and every vector $v$ can be represented as linear combination of elements of the semibasis ...

If elements in the set $SB$ are linearly "independent", and every vector in $V$ can be represented as a linear combination of elements in the set, then one can show that the representation must be unique. The official name for such set is "basis". If you don't require linear independence, then there are quite a few ways to describe such a set:

For expressing that a vector space $V$ is a span of a set $S$, one commonly uses the following phrases:

  • $S$ spans $V$;
  • $V$ is spanned by $S$;
  • $S$ is a spanning set of $V$;
  • $S$ is a generating set of $V$.

See more in this Wikipedia article: https://en.wikipedia.org/wiki/Linear_span

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