Guys Im having a little trouble understanding how and why we use complex vector spaces to describe the quantum states of a particle. Why complex vector spaces, and how is a complex vector space defined. Also are the 'vectors' in the field of quantum mechanics simply elements of a vector space like real numbers, or are the vectors actually analogous to actual vectors. Leonard susskind at Stanford already said that they are ONLY elements of a set of elements but i dont seem to understand how we can do STANDARD vector operations if they are not STANDARD vectors and are only ABSTRACT vectors. Can someone please help me with basic intuition behind the mathematics of quantum mechanics.