If "n" and "m" are both one, then the derivative is a number and the expression is the product of two numbers. But in higher dimensions, it is impossible for to be a number. If it were a number, then would be a vector in R"n" while the other terms would be vectors in R"m", and therefore the formula would not make sense. For the linear approximation formula to make sense, must be a function that sends vectors in R"n" to vectors in R"m", and must denote this function evaluated at v.