# Find a basis for the space of finitely nonzero sequences or real numbers

Let $$V$$ be the vector space of finitely nonzero sequences or real numbers. Find a basis $$B$$ of $$V$$.

My attempt I have to find a set $$B$$ such that: (i) $$\text{span}(B)=V$$ and (ii) if $$x\in B$$, then $$x\not\in \text{span}(B-\{x\})$$.

This set is very abstract and I have no idea where to start from. Any help or hint?

## 1 Answer

Let me give you some hints. The vector space $$V$$ has generic element $$x = (a_1,a_2,a_3,\ldots)$$, where $$a_i\in F$$ and $$a_i=0$$ for finitely many $$i$$. For instance, an element of $$V$$ is $$y = (0,0,1,2,0,0,\ldots)$$, with only two non-zero elements.

Now let $$B = \{e_0,e_1,e_2,\ldots\}$$, where $$e_0=(0,0,0,\ldots)$$, $$e_1=(1,0,0,0,\ldots)$$, $$e_2 = (0,1,0,0,\ldots)$$ and so on. Consider $$x\in V$$, with $$x = (a_1, a_2,a_3,\ldots)$$ a sequence with $$n$$ non-zero terms and let the nonzero position be $$i_1,\ldots,i_n$$.

For example, if $$x = (0,0,1,2,0,1.2,0\ldots)$$, then the non-zero terms are at the positions 2, 3 and 5, thus $$i_1=1, i_2=3,i_3=5$$.

Now check that properties (i) and (ii) hold.

• so in order to express $x$ as linear combination of the basis do I have to do something like $x = a_{i_1}e_{i_1}+a_{i_2}e_{i_2}+\cdots+a_{i_n}e_{i_n}$ ? Dec 7, 2022 at 8:44
• that's right! that shows the basis is generating... Dec 7, 2022 at 8:45
• thank you very very much!! Bad I don't have enough reps to upvote your anser... Dec 7, 2022 at 8:47