Mutually Singular Implicit Functions

A continuously differentiable (or smooth or in ′ (1)) function 1 from an open set which is a subset of a linear space X1 of dimension 2n into a linear space X2 of dimension n has an inverse onto a subspace upto equal dimension. If (s, t) is in where 1 (,) = , 1 ′ (,) = invertible at (s, t), then 1 is invertible in the neighbourhood of (s, t). Upto bijection, the invertibility of a multi variate function 1 is possible and for this 1 from X1 is mapped onto X2 so as to get invertible for the 1 st n components of the 2n-tuple. In the present discussion, we focus on two multi variate functions 1 and 2 or linear mappings from X1 into X2 such that both are smooth in two open sets 1 and 2 subsets of 1 taking the 1st n-components of 2n-tuples in X1 onto 2 while 2 is from the open set 2 of 1 taking the 2 nd set of n-components of 2n-tuples in X1 such that 1 ′ is invertible at (1 , 1) 1 and 2 ′ is invertible at (2 , 2) 2. The sum of derivative of the implicit functions is a continuously differentiable function gives 1 ′() + 2 ′() =-{ 1 −1 1ή + 2ή −1 2 } ...... 0.1 with 1 and 2 are the implicit functions, 1 = 1 + 1ή and 2 = 2 + 2ή in which 1 and 2ή are the invertible parts while 1ή and 2 are singular parts. Further, the null space of 1 is the range space of 2 and vice versa leading to 1 and 2 are mutually singular, denoted by 1┴ 2 ......0.2 Key words: linear space, multi variate function or a vector valued function, nonsingular mapping, mutually singular mapping, invertible mapping, null space (), range space (), dimension of a linear space, span, rank and nullity of a linear transformation, contraction, upto invertibility, subset, open mapping.