Chapter 3 Linear transformations

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(3) T: R 3 → R 3 , where T(u)=(-x, 2, 3) for each u=(x,y,z) in R 3 .2. Let T: R 3 → R 3 be defined by T(x,y,z)=(x,0,z). Find rank(T) and nullity(T).3. Let T and F be two linear transformations from V to W and k and l be scalars. Showthat kT+lF is also a linear transformation, where (kT+lF)(u)=kT(u)+lF(u) for every u inV. Show that all the linear transformations from V to Wform a vector space with respect to the addition and scalar multiplication of lineartransformations.4.Let M n× nbe the vector space of all n × n real matrices. Let A be a fixed n × nmatrix. Defines: M n× n→ M n× nby s (X)=AX-XA for each X in M n n.(1) Show that s is a linear transformation .(2) Prove for any X,Y in M n× n, s (XY)= s (X)Y+Xs (Y).5*. For any a, b in L(V), define [ a, b ]=ab - b a . Show that for any a , b,g in L(V),the following equation holds[[a , b ],g ]+[[ b ,g ],a ]+[[g ,a ], b ]=0,where ab is the composition of a and b .6. Let F: P 2 → P 2 be defined by F(ax 2 +bx+c)=(a-b)x 2 +(b+c).Show that F is a linear transformation. Find nullity(F) and rank(F).3.2 Matrices of linear transformationsThe linear transformation F: P 2 → P 2 given by F( ax 2 +bx+c)=(a+b)x+(b+c) is lessfamiliar than the transformation T: R 3 → R 2 defined by T(x,y,z)=(x+y, y+z).However F is uniquely determined by T. We can regard T as a concrete representationof the abstract transformation F.The matrix form of the linear transformation T(x,y,z)=(x+y,y+z) isTu=Au,where1 1 0⎡ ⎤A = ⎢ ⎥ .⎣01 1⎦
Also (1,0 )=T(1,0,0)=T(e 1 ), (1,1)=T(0,1,0)=T(e 2 ), (0,1)=T(0,0,1)=T(e 3 ) give the threecolumns of A, and { e 1 , e 2 , e 3 } is the standard basis of R 3 .3.2.1 Theorem Let T: R n → R m be a linear transformation and letB={ e 1,e 2,…,e n} be the standard basis of R n . Let A=[ T(e 1) T(e 2)…T(e n)] be them× n matrix with column vectors T(e i)(i=1,2,…,n). Then T=T A.The above matrix A determined by T is called the standard matrix of T.Exercise Let T: R 3 → R 2matrix of T.be defined by T(x,y,z)=(x, x+y,x+y+z). Find the standardNow let T: V → W be a linear transformation and B and S be a given basis of V and Wrespectively, then there exist linear isomorphisms[.] B: V→ R n and [.] S:W → R m ,here we assume V and W have dimension n and m respectively. Then the followingcomposition of three linear transformations is linear,R n −1[.] BV T W [.] SR m .By Theorem 3.2.1 there is a matrix A such that T Aequals this composition. Thismatrix A is uniquely determined by T and is called the matrix of T with respect to thebases B and S.2.2.2 Definition Let T: V → W be a linear transformation from the n-dimensional spaceV into m-dimensional space W, and let B and S be bases of V and W respectively. Theunique matrix A that makes the following diagram
Page 1 and 2: The Landmark Tshipi Acquisition & C
Page 3 and 4: Thus P 2 is isomorphic to R 3 , and
Page 5: ker(T) ={p(x): p’(x)=0}={ all con
Page 9 and 10: Thus [T]S, B=A=[ F(e 1) F(e 2) …
Page 11 and 12: Solution: By the above resultP=[[1]
Page 13 and 14: is invertible and A −1is similar

Chapter 3 Linear transformations

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