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A set $S$ in a vector space $V$ is linearly dependent if it contains finitely many linearly dependent vectors.

What is the significance of this definition? Obviously for finite sets, this is no different than the original definition. So what significance does it hold for infinite sets? I know it's not saying that infinite sets are always linearly independent, because that's not true (for example, the set of all real numbers, an infinite set, is linearly dependent).

Can you give me an example of a way in which this definition can be useful? Any help is appreciated!

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  • $\begingroup$ An example where the result may be counterintuitive: over the vector space of continuous functions over $[0,1]$, the functions $\{e^x,1,x,x^2,x^3,\dots\}$ are linearly independent. However, in a context where limits of sequences of functions are defined, we might say $$ e^x + \sum_{k=0}^\infty \frac {-1}{n!} x^n = 0 $$ i.e. there is a non-trivial infinite linear combination such that $\sum \alpha_i v_i = 0$ $\endgroup$ – Omnomnomnom Jan 19 at 19:47
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I suppose that wherever you saw this, linear dependence is defined in two steps:

  1. It is defined first what is the meaning of saying that a finite subset $S$ of a vector space $V$ is linearly dependent.
  2. Then, it is stated that an infinite subset $S$ of a vector space $V$ is linearly dependent if it contains a finite subset $S^\star$ which is linearly dependent.

In any case, a set $S$ is linearly dependent if and only of there elements $v_1,\ldots,v_n\in S$ and scalars $\lambda_1,\ldots,\lambda_n$, not all of which are $0$, such that $\lambda_1 v_1+\cdots+\lambda_nv_n=0$. This applies in both cases (that is, both when $S$ is finite and when $S$ is infinite).

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Recall that a finite set $\lbrace x_1,\dots ,x_n\rbrace\subset V$ (where $V$ is a vector space over some field $k$) is called linearly dependent if we can find $\alpha_1,\dots ,\alpha_n \in k$ with at least one $\alpha_i \neq 0$ such that $$ \sum_{i=1}^n \alpha_i x_i = 0. $$ On the other hand, this definition makes no sense for an infinite set since the sum of an infinite set of vectors is not defined (in an arbitrary vector space).

By defining a linearly dependent infinite set to be a set containing finitely many dependent vectors, infinite linearly independent sets retain the useful properties of their finite counterparts. For instance, any linearly independent set is contained in a basis (assuming Zorn's lemma).

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