We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Specifically, we employ recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of uncorrelated and semi-correlated random dual Wishart matrices of finite sizes, to design Hypothesis-Testing algorithms to detect the presence of PU signals. An inherent characteristic of the SCN/RMT-based approach, is that no signal-to-noise ratio (SNR) estimation or any other information on the PU signal is required. On top of this property, other attractive advantages of the new techniques are: a) due to the accuracy of the finite SCN distributions, superior performance is achieved under a finite number of samples, compared to asymptotic RMT-based alternatives; b) since expressions to model the SCN statistics both in the absence (H0) and presence (H1) of PU signal are used, the statistics of the spectrum sensing problem in question is completely characterized; c) as a consequence of a) and b), accurate and simple analytical expressions for the receiver operating characteristic (ROC) - both in terms of PD as a function of PF and in terms of PA as a function of PM - are yielded. It is also shown that the proposed finite RMT-based algorithms outperforms all similar alternatives currently known in the literature, at a substantially lower complexity.