When implemented in software (or hardware), a cryptographic protocol can leak sensitive information during its execution. Side-channel attacks can use those leakages in order to reveal some information about the secret used by the algorithm. The leaking side-channel information can take place in many time samples. Measurement appliances can cope with the acquisition of multiple samples. From an adversarial point of view, it is therefore beneficial to attempt to make the most of highly multivariate traces. On the one hand, template attacks have been introduced to deal with multivariate leakages, with as few assumptions as possible on the leakage model. On the other hand, many works have underlined the need for dimensionality reduction. In this paper, we clarify the relationship between template attacks in full space and in linear subspaces, in terms of success rate. In particular, we exhibit a clear mathematical expression for template attacks, which enables an efficient computation even on large dimensions such as several hundred of samples. It is noteworthy that both of PoI-based and PCA-based template attacks can straightforwardly benefit from our approach. Furthermore, we extend the approach to the masking-based protected implementations. Our approach is validated both by simulated and real-world traces.