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Suppose I have $N$ continuous independent random variables (random vectors) defined on $\mathbb{R}^N$. Can I comment on the probability of a particular realization of these N vectors being LINEARLY independent? In particular, can I comment that $P(\sum_i \alpha_i X_i=0) = 0$ for any set of $\alpha_i\in \mathbb{R}$ (expect all zeros) and where $X_i$'s are the $N$ random vectors.

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  • $\begingroup$ We need to know something about the distribution of this random vectors. With a Dirac delta distribution, it is also possible that such probability is one. $\endgroup$ – Jack D'Aurizio Sep 15 '14 at 9:51
  • $\begingroup$ Since they are continuous, Dirac's are excluded. $\endgroup$ – Ajay Sep 15 '14 at 11:31
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For a continuous distribution of $\mathbb{R}^n$, the measure of a $d$-dimensional vector space $V\subset\mathbb{R}^n$ with $d<n$ is zero, hence $$\mathbb{P}[X_n\in\operatorname{Span}(X_1,\ldots,X_{n-1})]=0,$$ so you are right.

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  • $\begingroup$ Thanks. I believe you meant $P[X_n \in \text{Span}(X_1,\cdots, X_{n-1})]=0$ $\endgroup$ – Ajay Sep 15 '14 at 14:04

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