Chapter 5 Linear Transformations and Operators

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Chapter 5Linear Transformations The Algebra of Linear TransformationsTheorem vector spaces over the fieldF. LetTandUbetwo Linear Transformations fromVintoW. The function(T+U)defined pointwiseby(T+U) (v) =Tv+Uvis a Linear transformation fromVintoW. Furthermore, ifs F, the function(sT)defined by(sT) (v) =s(Tv)is also a Linear transformation fromVintoW. The set of all Linear transformationfromVintoW, together with the addition and scalar multiplication defined above,is a vector space over the thatTandUare Linear transformation fromVintoW. For(T+U)defined above, we have(T+U) (sv+w) =T(sv+w) +U(sv+w)=s(Tv) +Tw+s(Uv) +Uw=s(Tv+Uv) + (Tw+Uw)=s(T+U)v+ (T+U)w,8384CHAPTER 5.

Linear Transformations and Operators 5.1 The Algebra of Linear Transformations Theorem 5.1.1. Let V and Wbe vector spaces over the field F. Let Tand Ube two linear transformations from Vinto W. The function (T+U) defined pointwise by (T+ U)(v) = Tv+ Uv is a linear transformation from Vinto W. Furthermore, if s2F, the function (sT) defined by ...

  Linear, Transformation, Linear transformations