Linear algebra doubt about the use of the word 'finite'

Let $V$ be a vector space and $S$ be a non empty subset of $V$. If $v\in\operatorname{Span}(S)$ then it is a linear combination of a 'finite' number of vectors in the subset $S$. What does the word finite signify? Could have just said the linear combination of the vectors of $S$.

You are correct that "linear combination" means a finite sum. The use of the word finite is for emphasis. For example you would not count the "infinite" linear combination

$$\exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!}$$

to be in the span of $\{1,x,x^2,\dots\}$ in the vector space $C[0,1]$ of continuous functions on $[0, 1]$.

• Why would I not consider it an infinite one, if I may ask? – user160370 Jun 28 '17 at 23:02
• @user160370 Sorry, that was poorly worded. Hopefully it's clearer now. – Trevor Gunn Jun 28 '17 at 23:09
• Yeah! Thanks :) – user160370 Jun 28 '17 at 23:11
• @T.Gunn could you please give a little more detail about why exp(x) is not in the span of the basis you suggest? – costrom Jun 29 '17 at 13:25
• @costrom I think you should make a new question to ask that. – Trevor Gunn Jun 29 '17 at 14:07

The only issue that might arise is how to define an infinite linear combination. After all, in Calculus, an infinite sum $$\sum_{n=1}^\infty a_n$$ is defined to be the limit $$\lim_{k\to \infty}\sum_{n=1}^ka_n.$$ This requires the notion of a limit, which in turn requires the notion of a topology, which our space $$V$$ need not have a priori. Thus, for introductory linear algebra (not assuming knowledge of topology) we restrict our attention to finite linear combinations.

It's worth pointing out that there can be vector spaces where some "infinite linear combinations" would make perfect sense.

Consider the vector space $\mathbb{R}^{\omega}$ without the restriction of finite support. This is a perfectly good vector space over $\mathbb{R}$ if you define addition coordinate-wise and multiplication by scalar you get a standard vector space.

In this vector space if you have any countable set of vectors $v_i$ and scalars $a_i$ that satisfy the condition that $\{i|\;j\in \mathrm{supp}(v_i)\}$ is finite for each $j$, then the $\sum_0^\infty a_i v_i$ is quite well defined and you could easily consider calling this a linear combination.

This is another reason to specifically call to attention the finiteness of the linear combination even though in most courses that is part of the definition.