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 P: 662 Hi, Everyone: I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension" a student asked me what was the use or purpose of a basis for a vector space V. All I could think of is that bases allow us to define a linear map L for all vectors, once we know the value of L at the basis vectors for V, i.e., vector spaces are free in their bases and so on. I mumbled something about identifying all vector spaces over the same field by their dimension, i.e., if V,V' v.spaces over F both, with the same dimension, then they are isomorphic. Are there other aspects where bases are equally important or more? Thanks.