In this paper we develop a hidden Markov model (HMM), called the linguistic HMM (LHMM), suitable for processing sequences of fuzzy vectors. A fuzzy vector B is an n-tuple of fuzzy numbers. Since fuzzy numbers are often associated with linguistic terms, such as "small," "medium," etc., a fuzzy vector can also be called a linguistic vector. The derivation of the linguistic HMM (LHMM) from the numeric HMM is done using the extension principle and the decomposition theorem. We show that the LHMM behaves in the same way as the HMM in the degenerate linguistic case when the fuzzy numbers are singletons (real numbers). We also derive the related algorithms for LHMM training (linguistic Baum-Welch) and for LHMM recognition (linguistic Viterbi). Several examples of LHMM training and recognition are given.