How to quantify the complexity of vectors whose components are real numbers? I started to study Kolmogorov complexity today, and  this question came to mind. Is there any way to use LZW to do this? I'm looking for a guidance to my studies
You can use LZW or other standard compression methods to approximate the Kolmogorov complexity. In some real-world cases this works well. However, it is dangerous, as it can also be a pretty bad approximation to the true Kolmogorov complexity, viz. http://bactra.org/notebooks/cep-gzip.html.

Another option is to seek the smallest Boolean circuit that outputs a given string. This problem appears to be hard*, but (a) isn't uncomputable like $K(x)$, and (b) doesn't suffer from the potentially erroneous measure given by gzip etc. (since circuits are a standard non-uniform, universal model of computation).

*[Update: It's actually essentially the Minimum Circuit Size Problem, which is looking like it's not $\mathsf{NP}$-hard, but it's still quite hard.]