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

Question

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 $S$.

Answer 1

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 $[0, 1]$.

Answer 2

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.

Answer 3

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.