Math 110 Fall 05 Lectures notes 11 Sep 23 Friday The next goal is to make explicit the connection between matrices familiar from Math 54 and linear transformations T V W between finite dimensional vectors spaces They are not quite the same because the matrix that represents T depends on the bases you choose to span V and W and the order of these bases Def Let V be a finite dimensional vector space An ordered basis of V is a basis for V with an order v 1 v n where n dim V Ex Let e i i th standard basis vector 1 in i th entry 0 elsewhere Then the bases e 1 e 2 e 3 and e 2 e 3 e 1 are the same order in a set does not matter but the ordered bases e 1 e 2 e 3 and e 2 e 3 e 1 are different Def For V F n e 1 e n is the standard ordered basis For P n F 1 x x 2 x n is the standard ordered basis Given ordered bases for V and W we can express both vectors in V and W and linear transformations T V W as vectors and matrices with respect to these ordered bases Def Let beta v 1 v n be an ordered basis for V For any x in V let x sum i 1 to n a i v i be the unique linear combination representing x The coordinate vector of x relative to beta denoted x beta is x beta a 1 a 2 a n ASK WAIT What is ASK WAIT Let V and v 3 6x x 3 ASK WAIT If beta v i beta P 5 F beta 1 x x 2 x 3 x 4 x 5 What is v beta x 5 x 4 x 3 x 2 x 1 Lemma The mapping Beta V F n that maps x to x beta is linear Proof if x sum i a i v i and y sum i b i v i then x beta a 1 a n y beta b 1 b n and 1 x y beta sum i a i b i v i beta by def of x y a 1 b 1 a n b n by def of beta a 1 a n b 1 b n x beta y beta by def of beta Similarly c x beta c x beta We need this representation of vectors in V as coordinate vectors of scalars in order to apply T V W as multiplication by a matrix We will also need to represent vectors in W the same way Let beta v 1 v n and gamma w 1 w m be ordered bases of V and W resp Let T V W be linear Then there are unique scalars a ij such that T v j sum i 1 to m a ij w i These scalars will be the entries of the matrix representing T Def Let T V W be linear V and W finite dimensional Using the above notation the m x n matrix A with entries a ij is the matrix representation of T in the ordered bases beta and gamma We write A T beta gamma If V W and beta gamma we write simply A T beta Note that column j of A is a 1j a mj T v j gamma To see why we call A the matrix representation of T let us use it to compute y T x Suppose x sum j 1 to n x j v j so x beta x 1 x n is the coordinate vector for x We claim the coordinate vector for y is just gotten by multiplying by A y gamma A x beta To confirm this we compute y T x T sum j 1 to n x j v j by def of x sum j 1 to n x j T v j since T is linear sum j 1 to n x j sum i 1 to m a ij w i by def of T v j sum j 1 to n sum i 1 to m a ij x j w i move x j into sum sum i 1 to m sum j 1 to n a ij x j w i reverse order of sums sum i 1 to m w i sum j 1 to n a ij x j pull w i out of inner sum 2 so y gamma sum j 1 to n a 1j x j sum j 1 to n a 2j x j sum j 1 to n a mj x j A x 1 A x beta x 2 as desired x n Ex T R 2 R 4 T x y x y 3 x 2 y 2 x 7 y beta standard basis for R 2 gamma standard basis for R 4 so T 1 0 1 3 2 0 and T 0 1 1 2 0 7 so A 1 1 for brevity in these notes we will sometimes use 3 2 Matlab notation T 1 1 3 2 2 0 0 7 2 0 0 7 ASK WAIT What if beta e2 e1 and gamma e3 e4 e1 e2 Ex continued Suppose x 3 e1 e2 what is T x what is T x gamma using standard bases T x T 3 1 4 7 6 7 T x gamma A 3 1 4 7 6 7 Ex T P 3 R P 2 R T f x f x beta 1 1 x x 2 x 3 gamma 2 x x 2 Then T 1 0 T 1 x 1 1 2 2 T x 2 2 x T x 3 3 x 2 So T 0 1 2 0 0 0 0 2 0 0 0 0 3 ASK WAIT What is T if beta 1 x x 2 x 3 If gamma 1 x x 2 Having identified matrices with linear transformations between two finite dimensional spaces with ordered bases and recalling that mxn matrices form a vector space we will not be surprised that all the linear transformations between any two vector spaces is also a vector space Def Let T and U be linear transformations from V W Then we define the new function T U V W by T U v T v U v and the new function c T V W by c T v c T v Thm Using this notation we have that 1 For all scalars c c T U is a linear transformation 2 The set of all linear transformation from V W 3 is itself a vector space using the above definitions of addition and multiplication by scalars Proof 1 c T U sum i a i v i c T sum i a i v i U sum i a i v i by def of c T sum i a i v i U sum i a i …