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Invariants play an important role in continuum mechanics. Knowing the number of independent invariants is crucial in modelling and in a rigorous construction of a constitutive equation for a particular material, where it is determined by doing tests that hold all, except one, of the independent invariants constant so that the dependence in the one invariant can be identified. Hence, the aim of this paper is to prove that the number of independent invariants for a set of n symmetric tensors and m unit vectors is at most 2m+6n-3. The prove requires the construction of spectral invariants. All classical invariants can be explicitly expressed in terms spectral invariants. We show that the number of spectral invariants in an irreducible functional basis is reduced to 2m+6n-3; a significant reduction to that obtained in the literature if the value of m or n is large. Relations between classical invariants in a classical-minimal integrity basis are given. [ABSTRACT FROM AUTHOR]