Fuzzy linguistic decision making methods are hot in decision making. Compared with the decision matrixes which are widely used in fuzzy linguistic decision making methods, alternatives-linguistic terms decision matrixes are provided to represent decision results of experts with respect to attributes, and the advantage of alternatives-linguistic terms decision matrixes in distinguishing linguistic decision making of distinct, partial unknown or hesitant linguistic environments is analyzed. Inspired by the classical TOPSIS decision method and vector operation, the positive and negative ideal solutions based on alternatives-linguistic terms decision matrixes are presented, pseudo-distances and relative closeness between the positive(negative) ideal solution and alternatives are calculated, and their properties are analyzed. Then based on the 2-tuple linguistic represent model, the fuzzy linguistic TOPSIS decision making method based on alternatives-linguistic terms decision matrixes are proposed. An example is given to illustrate the practicality of the proposed method. Compared with three existed important fuzzy linguistic decision making methods, the proposed method can be used to overcome the drawback of the existed method, which is an alternative fuzzy linguistic decision making method.