Definition of linear combination for an infinite subset $G$ of a vector space $<V,+,·>$ over a field $K$.

Question

My book offers two definitions of linear combination:

• Let $<V,+,·>$ be a vector space over a field $K$, and $G = \{ v_1, ..., v_r \} ⊆ V$. A Linear Combination of $G$ is an element $v∈V$ such that $v=\sum_{i=1}^r α_i · v_i$ with $α_i∈K$ for every $1≤i≤r$.

I get this one, but then it offers a "more general" one that says:

• Let $<V,+,·>$ be a vector space over a field $K$, $I$ a set of indexes and $G = \{v_i|i∈I\} ⊂ V$. A Linear Combination of $G$ is an element $v∈V$ such that $v=\sum_{i∈I} α_i · v_i$ where $α_i=0$, except for finite $i∈I$.

I don't get this one, and I think it's because I don't understand the role of the set $I$, also I have never used $\sum_{i∈I}α_i$ notation, just the regular one as in $\sum_{i=1}^r α_i · v_i$. In addition, I don't understand if when G is an infinite set then all $α_i$ have to be equal to $0$, or just some of them.

Answer 1

$\sum_{i∈I}α_i$ is defined naturally. Let's say $I=\{\text{Alice},\text{Bob},\text{Charlie}\}$. Then $$\sum_{i∈I}α_i = \alpha_{\text{Alice}} + \alpha_{\text{Bob}} + \alpha_{\text{Charlie}}$$

The only important thing is that when $|I| = \infty$, you require that only finitely many $\alpha_i$'s can be non-zero because addition of an infinite number of vectors in a vector space is undefined (unless there's additional structure like a topology on it. But for a vector space without any additional structure, it's undefined).

For example, polynomials are generated by the set $\{1,x,x^2,\cdots,x^n,\cdots\}$. So, the set of polynomials consists of all linear combinations $c_o\cdot 1+ c_1\cdot x + c_2\cdot x^2 + \cdots$ such that except finitely many of the coefficients, the rest are zero. So, the formal power series for $e^x$ is not a polynomial for example.

Answer 2

The second is for the scenario when you have an infinite basis. Take example of space of all polynomials with real coefficients. Any polynomial is going to be a linear combination of $ 1, x, x^2, x^3, ... $. But this basis is infinite, and all but finitely many of the coefficients will be zero. For example, $ x^2 + 2x + 4 = 4 \cdot 1 + 2 \cdot x + 1 \cdot x^2 + 0 \cdot x^3 + 0 \cdot x^4 + ... $.

Answer 3

A linear combination is a finite combination of vectors that have been rescaled. In the first example you have, the set of vectors you draw from is already finite so you don’t need to mention ‘finite’. In the second example, this set is infinite and so you need to mention the qualifier ‘finite’ that was elided in the first example.