Math 110 - Fall 05 - Lectures notes # 11 - Sep 23 (Friday)The next goal is to make explicit the connection betweenmatrices, familiar from Math 54, and linear transformationsT: V -> W between finite dimensional vectors spaces.They are not quite the same, because the matrix that representsT depends on the bases you choose to span V and W, andthe order of these bases:Def: Let V be a finite dimensional vector space. Anordered 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 vectorsin V and W, and linear transformations T:V->W as vectors andmatrices 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 uniquelinear combination representing x. The coordinate vectorof x relative to beta, denoted [x]_beta, is[x]_beta = [ a_1 ][ a_2 ][ ... ][ a_n ]ASK & WAIT: What is [v_i]_beta ?ASK & WAIT: Let V = P_5(F), beta = {1,x,x^2,x^3,x^4,x^5},and v = 3-6x+x^3. What is [v]_beta?ASK & WAIT: If 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] and1[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 []_betaSimilarly, [c*x]_beta = c*[x]_betaWe need this representation of vectors in V ascoordinate vectors of scalars in order to apply T: V -> Was multiplication by a matrix. We will also need torepresent 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 thatT(v_j) = sum_{i=1 to m} a_{ij}*w_iThese 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 withentries a_{ij}, is the matrix representation of T in the orderedbases beta and gamma. We write A = [T]_beta^gamma.If V = W and beta = gamma, we write simply A = [T]_betaNote that column j of A is [a_{1j};...;a_{mj] = [T(v_j)]_gammaTo see why we call A the matrix representation of T, let us useit 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 coordinatevector for y is just gotten by multiplying by A:[y]_gamma = A * [x]_betaTo 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 sum2so[y]_gamma = [ sum_{j=1 to n} a_{1j}*x_j ] = A * [ x_1 ] = A*[x]_beta[ sum_{j=1 to n} a_{2j}*x_j ] [ x_2 ] as desired[ ... ] [ ... ][ sum_{j=1 to n} a_{mj}*x_j ] [ 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), soA = [ 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^2SoT=[01/200][0020][0003]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 betweentwo finite dimensional spaces with ordered bases, and recallingthat mxn matrices form a vector space, we will not be surprisedthat all the linear transformations between any two vectorspaces 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,3is itself a vector space, usingthe above definitions of addition and multiplication by scalarsProof:(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+U= c*(T(sum_i a_i*v_i)) + U(sum_i a_i*v_i) ... by def of c*T= c*(sum_i a_i*T(v_i)) + sum_i a_i*U(v_i) ... since T,U linear= sum_i a_i*c*T(v_i) + sum_i a_i*U(v_i)= sum_i a_i*(c*T(v_i)+U(v_i))= sum_i a_i*(c*T+U)(v_i) ... by def of c*T+U(2) We let T_0, defined by T_0(v) = 0_W for all v, be the"zero vector" in L(V,W). It is easy to see that all theaxioms of a vector space are satisfied. (homework!)Def: L(V,W) is the vector space of all linear transformationsfrom V -> W. If V=W, we write L(V) for short.Given ordered bases for finite dimensional V and W, we get amatrix [T]_beta^gamma for every T in L(V,W). It is naturalto expect that the operations of adding vectors in L(V,W)(adding linear transformations) should be the same as addingtheir matrices, and that multiplying a vector in L(V,W) by ascalar should be the same as multiplying its matrix by a scalar:Thm: Let V and W be finite dimensional vectors spaces withordered bases beta and gamma, resp. Let T and U be in L(V,W).Then(1) [T+U]_beta^gamma = [T]_beta^gamma + [U]_beta^gamma(2) [c*T]_beta^gamma = c*[T]_beta^gammaIn other words, the function []_beta^gamma: L(V,W) -> M_{m x n}(F)is a linear transformation.Proof: (1) We compute column j of matrices on both sidesand comfirm they are the same. Let beta = {v_1,...,v_n}and gamma = {w_1,...,w_m}. Then(T+U)(v_j) = T(v_j) + U(v_j) so[(T+U)(v_j)]_gamma = [T(v_j)]_gamma + [U(v_j)]_gammaby the above Lemma that shows the mapping x -> [x]_gammawas