linear map(Or "linear transformation") A {function} from a {vector space} to a vector space which respects the {additive} and {multiplicative} structures of the two: that is, for any two {vectors}, u, v, in the source vector space and any scalar, k, in the {field} over which it is a vector space, a linear map f satisfies f(u+kv) = f(u) + kf(v). (1996-09-30)