Linear Transformations Chapter 2 Linear Transformations and Matrices Definition We call a function T V W a linear transformation from V to W if for all x y V and c F we have a T x y T x T y and b T cx cT x Per Olof Persson persson berkeley edu Department of Mathematics University of California Berkeley 1 If T is linear then T 0 0 2 T is linear T cx y cT x T y x y V c F 3 4 Math 110 Linear Algebra If T is linear then T x y T x T y x y V T isPlinear for Px1 xn V and a1 an F T ni 1 ai xi ni 1 ai T xi Special linear transformations The identity transformation IV V V IV x x x V The zero transformation T0 V W T0 x 0 x V Null Space and Range Definition For linear T V W the null space or kernel N T of T is the set of all x V such that T x 0 N T x V T x 0 The range or image R T of T is the subset of W consisting of all images of vectors in V R T T x x V Theorem 2 1 For vector spaces V W and linear T V W N T and R T are subspaces of V and W respectively Theorem 2 2 For vector spaces V W and linear T V W if v1 vn is a basis for V then Nullity and Rank Definition For vector spaces V W and linear T V W if N T and R T are finite dimensional the nullity and the rank of T are the dimensions of N T and R T respectively Theorem 2 3 Dimension Theorem For vector spaces V W and linear T V W if V is finite dimensional then nullity T rank T dim V R T span T span T v1 T vn Properties of Linear Transformations Theorem 2 4 For vector spaces V W and linear T V W T is one to one if and only if N T 0 Theorem 2 5 For vector spaces V W of equal finite dimension and linear T V W the following are equivalent a T is one to one b T is onto c rank T dim V Linear Transformations and Bases Theorem 2 6 Let V W be vector spaces over F and v1 vn a basis for V For w1 wn in W there exist exactly one linear transformation T V W such that T vi wi for 1 n Corollary Suppose v1 vn is a finite basis for V then if U T V W are linear and U vi T vi for i 1 n then U T Coordinate Vectors Matrix Representations Definition For a finite dimensional vector space V an ordered basis for V is a basis for V with a specific order In other words it is a finite sequence of linearly independent vectors in V that generates V Definition Let u1 un be an ordered basis for V and for x V let a1 an be the unique scalars such that n X x ai ui i 1 The coordinate vector of x relative to is a1 x an Addition and Scalar Multiplication Definition Let T U V W be arbitrary functions of vector spaces V W over F Then T U aT V W are defined by T U x T x U x and aT x aT x respectively for all x V and a F Theorem 2 7 With the operations defined above for vector spaces V W over F and linear T U V W a aT U is linear for all a F Definition Suppose V W are finite dimensional vector spaces with ordered bases v1 vn w1 wm For linear T V W there are unique scalars aij F such that T vj m X for 1 j n aij wi i 1 The m n matrix A defined by Aij aij is the matrix representation of T in the ordered bases and written A T If V W and then A T Note that the jth column of A is T vj and if U T for linear U V W then U T Matrix Representations Theorem 2 8 For finite dimensional vector spaces V W with ordered bases and linear transformations T U V W a T U T U b aT a T for all scalars a b The collection of all linear transformations from V to W is a vector space over F Definition For vector spaces V W over F the vector space of all linear transformations from V into W is denoted by L V W or just L V if V W Composition of Linear Transformations Theorem 2 9 Let V W Z be vector spaces over a field F and T V W U W Z be linear Then UT V Z is linear Theorem 2 10 Let V be a vector space and T U1 U2 L V Then a T U1 U2 TU1 TU2 and U1 U2 T U1 T U2 T b T U1 U2 TU1 U2 c TI IT T d a U1 U2 aU1 U2 U1 aU2 for all scalars a Matrix Multiplication Let T V W U W Z be linear v1 vn w1 wm z1 zp ordered bases for U W Z and A U B T Consider UT m m X X UT vj U T vj U Bkj wk Bkj U wk k 1 m X Bkj k 1 p X i 1 Aik zi k 1 p m X X i 1 k 1 Aik Bkj zi Definition Let A B be m n n p matrices The product AB is the m p matrix with n X AB ij Aik Bkj for 1 i m 1 j p k 1 Matrix Multiplication Properties Theorem 2 11 Let V W Z be finite dimensional vector spaces with ordered bases and T V W U W Z be linear Then UT U T Corollary Let V be a finite dimensional vector space with ordered basis and T U L V Then UT U T Definition The Kronecker delta is defined by ij 1 if i j and ij 0 if i 6 j The n n identity matrix In is defined by In ij ij Theorem 2 12 Let A be m n matrix B C be n p matrices and D E be q m matrices Then a A B C AB AC and D E A DA EA b a AB aA B A aB for any scalar a c Im A A AIn d If V is an n dimensional vector space with ordered basis then IV In Corollary Let A be m n matrix B1 Bk be n p matrices C1 Ck be q m matrices and a1 ak be scalars Then k k k k X X X X A ai Bi ai ABi and a i Ci …