Question

Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S.

Answer 1

Hint -Let V and w be finite dimensional vector spaces, and T:V---> W a

linear transformation. We denote by So W the range space of T, by 2 a

direct complement of ~4 , by X the kernel (or the null

space) of T and by & a direct complement o The

range space of any general transformation T will be indicated by R( T ). The

projection operator on ~2 along y is denoted by , and that on _,along These projection operators are well definedIf T: V--> w is not bijective, there is no unique inverse transformation

T: W ---> V. In such a case, an inverse can be defined only in some special

Reqson+ Take SVDO singulast value decomposition, dimensional space, let V be W be m an dimensional space. myn m be well Sit. v non OF A Uw Then T:Vw can be treated matrik let the matrix Comoresponding to ordered Basic OF V and w Then. by the Known singulay Valve Decomposition 3 unitary matrix U and ป m- U [...] OF zero AD's Diagonal mat olun eigen value Now define va [olo u Then monom = [8]" v° [ 3 ] u” [ 37v 4[BE]" and u and u are unctuary Similarly Nomon = matrica. [006] 4*0 [80] w posto ve 78:]u*=N Now let S be the lineau operator correskin to the matrix N. i's generalined inver OF T. NOTE isomorphic then it exist IF unigune = I=TOT To (TOT) = TOI ET T'o (TOT") = TM generalized inverne oft Siwmv sit be linear map inverne oFT generalished TIS Invertible S(Ts) = 5 SET! TI uniqueny gener alred inverse OFT S T is an Toti 15 a S is TS=I is

sense and for specific purposes. Early attempts at defining such inverses in

the case of a matrix transformation