Question

Let V and W be finite dimensional vector spaces and let T:V → W be a linear transformation. We say a linear transformation S :W → V is a left inverse of T if ST = Iy, where Iy denotes the identity transformation on V. We say a linear transformation S:W → V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S:W → V is an inverse of T if it is both a left and right inverse of T. When T has an inverse, we say T is invertible. Show that (a) T has a right inverse iff T is surjective. (b) If a is a basis for V and ß is a basis for W, then [T]. has a right inverse iff its rows are linearly independent.

Answer 1

Aight invesse implies Ras aT: V W uck that T.S ToS (w)I () Tis sujective Comvercely suppose T is sujective Rese T)o Now let w W Sine ais DeAne wRese T ,T So Hat T+=2 T)-T) cw limeos mab quet that hen (o,+ = v, +V2 and o)-ev SW V is T (u) w T) Whele To S(4) To S-I

Let dim V- h and dim Wm ITJ=A en mxn motsix Let A is an Let B Pright invesse A. be nm A Let Rbe tRe i how AB Corider s ase scas C 7(C = 0 -AB m Incady indepandt CImplies ows Lr oue Convessely Suppose mdependent. Since A is mxn coDsides A core linecgly RONS matsix mn T an nx m mathix oRank Now T Reduced AoN ecfelon fom A' is Hence Einveltible matix suck TRat hxh (onride le poshtion Now EA = O n-mxm T (p T B צמווXמ BA I So tat EA= n-mxh A Ras Right invesse. AB