The Kochen-Specker theorem is one of the fundamental theorems in non-contextuality hypothesis that assumes that the values of observables are independent of the context, i.e., of the way that measurements are performed on the system and therefore predetermined, does not hold for quantum systems. The proof is provided by any counterexample to the assumed pre-existence of values of observables in quantum mechanics (such as the spin of a system). The theorem not only characterizes quantum systems but is also one of the major arguments against hidden variables theories that assume that the ambiguity of the measurements of observables may be ascribed to hidden variables that are not measured. We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of space, $\calH^n$, $n\ge3$ to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate linear MMP diagrams corresponding to blocks of orthogonal vectors in $\mathbbR^n$, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of interval analysis. %statiscally polynomially complex interval analysis and %self-teaching programs. The algorithms are limited neither by the %number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors are found.