Chapter 4 Linear Transformations In this chapter, we introduce the general concept of linear transformation from a vector space into a vector space. But, we mainly focus on linear transformations from n R to m R.§1 Definition and ExamplesNew words and phrasesMapping 映射Linear transformation 线性变换Linear operator 线性算子Dilation 扩张Contraction 收缩Projection 投影Reflection 反射Counterclockwise direction 反时针方向Clockwise direction 顺时针方向Image 像Kernel 核1.1 Definition★Definition A mapping(映射) L: V W is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set.★Definition A mapping L from a vector space V into a vector space W is said to be a linear transformation（线性变换）if(1) 11221122(v v )(v )(v )L L L αααα+=+for all 12v ,v V ∈ and for all scalars 1α and 2α. (1) is equivalent to(2) 1212(v v )(v )(v )L L L +=+ for any 12v ,v V ∈ and(3) (v)(v)L L αα= for any v V ∈ and scalar α.Notation: A mapping L from a vector space V into a vector space W is denotedL: V →WWhen W and V are the same vector space, we will refer to a linear transformation L: V →V as a linear operator on V . Thus a linear operator is a linear transformation that maps a vector space V into itself.1.2 Linear Operators on 2R1. Dilations(扩张) and Contractions Let L be the operator defined byL(x)=k xthen this is a linear operator. If k is a positive scalar, then the linear operator can be thought of as a stretching or shrinking by a factor of k.120(x)x 0x k L A x k ⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭2. Projection （投影）onto the coordinate axes.L(x)=11e x 1210(x)x 00x L A x ⎛⎫⎛⎫== ⎪⎪⎝⎭⎝⎭ L(x)=22e x 1200(x)x 01x L A x ⎛⎫⎛⎫== ⎪⎪⎝⎭⎝⎭3. Reflections (反射) about an axis Let L be the operator defined byL(x)=12(,)T x x -, then it is a linear operator. The operator L has theeffect of reflecting vectors about the x-axis. 1210(x)x 01x L A x ⎛⎫⎛⎫== ⎪ ⎪-⎝⎭⎝⎭Reflecting about the y-axis L(x)=12(,)T x x -, 1210(x)x 01x L A x -⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭4. RotationsL(x)=21(,)T x x -, L has the effect of rotating each vector by 90 degrees in the counterclockwise direction （逆时针方向）.1201(x)x 10x L A x -⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭1.3 Linear Transformations from n R to m RIf A is an mxn matrix, then we can define a linear transformation A L from n R to m R by()A L X AX =It is easy to verify that the mapping above is linear. In the next section, we will see that any linear transformation from n R to m R must be of this form.1.4 The Image and Kernel★Definition Let L: n R →m R is a linear transformation. The kernel （核）of L denoted ker(L), is defined by ker(L)={}v |(v)0W V L ∈=★Definition Let L: n R →m R is a linear transformation and let S be a subspace of V . The image （像）of S, denoted L(S), is defined by L(S)= {}w |w (v) for some v m n R L R ∈=∈The image of the entire vector space, L(V), is called the range （值域）of L.Theorem 4.1.1 If L: n R →m R is a linear transformation and S is a subspace of n R , then (i) ker(L) is a subspace of n R . (ii) L(S) is a subspace of m R .Assignment for section 1, chapter 4Hand in: 3, 4, 17, 20,Not required : 8, 10, 11, 15, 16, 19, 25§2 Matrix Representations of Linear TransformationsNew words and phrasesMatrix representation 矩阵表示 Formal multiplication 形式乘法 Similarity 相似性2.1 Matrix Representation of Linear TransformationsIn section 1 of this chapter, the examples of linear transformations can be represented by matrices. In general, a linear transformation can be represented by a matrix.If we use the basis E=[12u ,u ,,u n ] for U and the basisF=[12v ,v ,,v m ] for V , and L: U → V .If u is a vector in U, then1122u u u u n n x x x =+++ (in U) |→ 1122L(u)v v v m m y y y =+++ (in V)The linear transformation L is determined by the change of the coordinate vectors:12n x x x ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭→12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭Assume that1122(u )v v v j j j mj m L a a a =+++, j=1, 2, …, nFormally,12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪⎪⎪⎝⎭Write 12[(u ),(u ),,(u )]n L L L as linear combinations of 12[v ,v ,,v ]m , then consequently, A is obtained.Then L(u) = 1122(u u u )n n L x x x +++ =12[v ,v ,,v ]m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭(formal multiplication)1122L(u )L(u )L(u )n n x x x =+++=nj=1(u )j j x L ∑1212[L(u ),L(u ),,L(u )]n n x xx ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭(formal multiplication)==11(v )n mj ij i j i x a =∑∑=i=11()v m nij j i j a x =∑∑=12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪ ⎪⎪⎝⎭12n x x x ⎛⎫ ⎪ ⎪⎪ ⎪⎝⎭Hence12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭=111212122212n n m m mn a a a aa a a a a ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭12n x x x ⎛⎫ ⎪ ⎪⎪ ⎪⎝⎭Thus, y=A x is the coordinate vector of L (u) with respect toF =[12v ,v ,,v m ]. y=A x is called the matrix representation of thelinear transformation. A is called the matrix representing L relative to thebases E and F. A is determined by the following equations.12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪⎪⎪⎝⎭We have established the following theorem. Theorem 4.2.2 If E=[12u ,u ,,u n ] is an ordered basis for U andF=[12v ,v ,,v m ] is an ordered basis for V , then corresponding to eachlinear transformation L:U →V there is an mxn matrix A such that [()][]F E L u A u = for each u in U.A is the matrix representing L relative to the ordered bases E and F. In fact, a [(u )]j j F L =.2.2 Matrix Representation of L: n R →m RIf U=n R , V=m R , then we have the following theorem.Theorem 4.2.1 If L is a linear transformation mapping n R into m R , there is an mxn matrix A such thatL (x)=A xfor each x n R ∈. In fact, the jth column vector of A is given by12((e ),(e ),,(e ))n A L L L =Proof If we choose standard basis 12[e ,e ,,e ]n for n R and thestandard basis 12[e ,e ,,e ]m for m R ,L(x)= 1122(e e e )n n L x x x +++=12(e ,e ,,e )m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭=12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭1122L(e )L(e )L(e )n n x x x =+++=nj=1(e )j j x L ∑1212(L(e ),L(e ),,L(e ))n n x x x ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭And let A=()ij a =()12a ,a ,,a n 12(L(e ),L(e ),,L(e ))n =If 1122x e e e n n x x x =+++, then L(x)=Ax.A is referred to as the standard matrix representation (标准矩阵表示) of L.( A representation with respect to the standard basis.)Example 1 (example 1 on page 186) Determine the standard matrix representation of L.Define the linear transformation L:3R 2R by1223(x)(,)T L x x x x =++ for each 123x (,,)T x x x = in 3R , find the linear standard representation of L.Solution: Find 123L(e ),L(e ),L(e ). Then 123110(L(e ),L(e ),L(e ))011A ⎛⎫== ⎪⎝⎭Example 2 rotation by an angle θLet L be the linear transformation operator on 2R that rotates each vector by an angle θ in the counterclockwise direction. We can see that1e is mapped to (cos ,sin )T θθ, and 2e is mapped to (sin ,cos )T θθ-.1(e )(cos ,sin )T L θθ=, 2(e )(sin ,cos )T L θθ=-The matrix A representing the transformation will be12cos sin (L(e ),L(e ))sin cos A θθθθ-⎛⎫==⎪⎝⎭To find the matrix representation A for a linear transformation L n R→ m R w.r.t. the bases E=[12u ,u ,,u n ] and F=[12b ,b ,,b m ], wemust represent each vector 1122(u )b b b j j j mj m L a a a =+++. The followingtheorem shows that determining this representation is equivalent to solving the linear system Bx=(u )j L , where (u )j L is regarded as a column vector in m R .Theorem 4.2.3 Let E =[12u ,u ,,u n ] and F =[12b ,b ,,b m ] beordered bases for n R and m R , respectively. If L : n R → m R is a linear transformation and A is the matrix representing L with respect to E and F , then112((u ),(u ),,(u ))n A B L L L -=where B =(12b ,b ,,b m ).Proof L(u) = 1122(u u u )n n L x x x +++ =12(b ,b ,,b )m 12m y y y ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭1122L(u )L(u )L(u )n n x x x =+++=nj=1(u )j j x L ∑1212(L(u ),L(u ),,L(u ))n n x xx ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭12(b ,b ,,b )m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭1212(L(u ),L(u ),,L(u ))n n x x x ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭The matrix B is nonsingular since its column vectors form a basis form R . Hence, 112((u ),(u ),,(u ))n A B L L L -=12((u ),(u ),,(u ))n L L L is the matrix representing L relative to the bases [12u ,u ,,u n ] and [12e ,e ,,e m ]. B is the transition matrixcorresponding to the change of basis from [12b ,b ,,b m ] to[12e ,e ,,e m ].Corollary 4.2.4 If A is the matrix representing the linear transformation L:n R m R with respect to the bases12m [b ,b ,,b ]1B -12m [e ,e ,,e ]12n [u ,u ,,u ] A12((),(),,())n L u L u L uBE=[12u ,u ,,u n ] and F=[12b ,b ,,b m ]then the reduced row echelon form of1212(b ,b ,,b |(u ),L(u ),,L(u ))m n Lis (I |A )Proof 1212(b ,b ,,b |(u ),L(u ),,L(u ))m n L =(B|BA), which is rowequivalent to (I|A).Examples Finding the matrix representing L Example 3 on page 188Let L be a linear transformation mapping 3R into 2R defined by 11232L(x)b ()b x x x =++. Find the matrix A representing L with respect to the ordered bases 123[e ,e ,e ] and 12[b ,b ], where 11b 1⎛⎫= ⎪⎝⎭, 21b 1-⎛⎫= ⎪⎝⎭Solution:Method 1. Represent 123[e ,e ,e ] in terms of 12[b ,b ] Method 2. 112123(b ,b )((e ),(e ),(e ))A L L L -=1111111/21/2111100111111/21/2111011A ------⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫=== ⎪ ⎪ ⎪⎪ ⎪-⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭Method 3 Applying row operations. 12123(b ,b |(e ),(e ),(e ))L L L Example 4 on page 188Let L be a linear transformation mapping 2R into itself defined by 1212L(b b )()b 2b αβαββ+=++, where 12[b ,b ] is the ordered basisdefined in example 3. Find the matrix A representing L with respect to12[b ,b ].Solution: Use three methods as in example 3. Example 6 on page 190Determine the matrix representation of L with respect to the given bases. Let L: 2R 3R be the linear transformation defined by 21212L(x)(,,)T x x x x x =+-Find the matrix representation of L with respect to the ordered bases12[u ,u ] and 123[b ,b ,b ], where12u (1,2),u (3,1)T T ==123b (1,0,0),b (1,1,0),b (1,1,1)T T T ===Assignment for section 2, chapter 4Hand in: 2, 6, 8, 16, 20 Not required: 9—15, 17, 19§3 SimilarityLet L be a linear operator on V , E=[12v ,v ,,v n ] be an orderedbasis for V , A is the matrix representing L with respect to the basis E. 1122u v v v n n x x x =+++, 1122L(u)v v v n n y y y =+++1211221212(v )v v v [v ,v ,,v ][v ,v ,,v ]a j j j j j nj n n n j nj a a L a a a a ⎛⎫ ⎪ ⎪=+++== ⎪ ⎪ ⎪⎝⎭y=Ax F=[12w ,w ,,w n ]1122u w w w n n c c c =+++, 1122L(u)w w w n n d d d =+++1211221212(w )w w w [w ,w ,,w ][w ,w ,,w ]b j jj j j nj n n n j nj b b L b b b b ⎛⎫ ⎪ ⎪=+++== ⎪ ⎪ ⎪⎝⎭d=BcLet the transition matrix corresponding the change of basis from F=[12w ,w ,,w n ] to [12v ,v ,,v n ]Then x=Sc, y=Sd, 11y x S BS --= 1y x SBS -= or 1A SBS -=Hence, we have established the following theorem.Theorem 4.3.1 Let E=[12v ,v ,,v n ] and F=[12w ,w ,,w n ] betwo ordered bases for a vector space V , and let L be a linear operator onn R . Let S be the transition matrix representing the change from F to E. IfA is the matrix representing L with respect to E, andB is the matrix representing L with respect to F, then 1B S AS -=.★Definition Let A and B be nxn matrices. B is said to be similar to A if there is a nonsingular matrix S such that 1B S AS -=.Example 2 (on page 204)Example Let L be the linear operator on 3R defined by L(x)=Ax, whereV VV VBasis E=12n [v ,v ,,v ] Su →L(u)u →L(u)Ax=yBc=dS -1Coordinate vector of L(u): dBasis FCoordinate vector of L(u) :y Basis ECoordinate vector of u: xCoordinate vector of u: c Basis E=12n [w ,w ,,w ] x=Sc y=Sd220112112⎛⎫ ⎪ ⎪ ⎪⎝⎭. Thus the matrix A represents L with respect to the standard basis for 3R . Find the matrix representing L with respect to the basis [123y ,y ,y ], where 1y (1,1,0)T =-, 2y (2,1,1)T =-, 3y (1,1,1)T =. SolutionD=000010004⎛⎫⎪⎪ ⎪⎝⎭is the matrix representing L w.r.t the basis [123y ,y ,y ],. Or, we can find D using 1D Y AY -= 11x ()x=()x n n n A YDY YD Y --=Using this example to show that it is desirable to find as simple as a representation as possible for a linear operator. In particular, if the operator can be represented by a diagonal matrix, this is usually preferred representation. It makes the computation of Dx and x n D easier.Assignment for section 3, chapter 4Hand in: 2, 3, 4, 8, 10, 15 Not required 5, 6。