Linear Transformations 6.1 Introduction to Linear Transformations 6.2 The Kernel and Range of a Linear Transformation 6.3 Matrices for Linear Transformations 6.4 Transition Matrices and Similarity 6.5 Applications of Linear Transformations 6.1 6.1 Introduction to Linear Transformations A function T that maps a vector space V into a vector space W: TVWVW:mapping , , : vector spaces V: the domain of T W: the codomain of T Image of v under T : If v is a vector in V and w is a vector in W such that T(),vw then w is called the image of v under T (For each v, there is only one w) The range of T : The set of all images of vectors in V (see the figure on the next slide) 6.2 The preimage of w: The set of all v in V such that T(v)=w (For each w, v may not be unique) The graphical representations of the domain, codomain, and range ※ For example, V is R3, W is R3, and T is the orthogonal projection of any vector (x, y, z) onto the xy-plane, i.e. T(x, y, z) = (x, y, 0) (we will use the above example many times to explain abstract notions) ※ Then the domain is R3, the codomain is R3, and the range is xy-plane (a subspace of the codomian R3) ※ (2, 1, 0) is the image of (2, 1, 3) ※ The preimage of (2, 1, 0) is (2, 1, s), where s is any real number 6.3 2 2 Ex 1: A function from R into R 2 2 2 T : R R v (v1,v2 )R T(v1,v2 ) (v1 v2 ,v1 2v2 ) (a) Find the image of v=(-1,2) (b) Find the preimage of w=(-1,11) Sol: (a) v ( 1, 2) TT(v ) ( 1, 2) ( 1 2, 1 2(2)) ( 3, 3) (b) T(vw ) ( 1, 11) T(v1,v2 ) (v1 v2 ,v1 2v2 ) (1, 11) v1 v2 1 v1 2v2 11 v1 3, v2 4 Thus {(3, 4)} is the preimage of w=(-1, 11) 6.4 Linear Transformation : V,W： vector spaces T :V W： A linear transformation of V into W if the following two properties are true (1) T(u v) T(u) T(v), u, vV (2) T(cu) cT (u), cR 6.5 Notes: (1) A linear transformation is said to be operation preserving (because the same result occurs whether the operations of addition and scalar multiplication are performed before or after the linear transformation is applied) T(u v) T(u) T(v) T(cu) cT (u) Addition Addition Scalar Scalar in V in W multiplication multiplication in V in W (2) A linear transformation T : V V from a vector space into itself is called a linear operator 6.6 2 2 Ex 2: Verifying a linear transformation T from R into R T(v1,v2 ) (v1 v2 ,v1 2v2 ) Pf: 2 u (u1,u2 ), v (v1,v2 ) : vector in R , c: any real number (1) Vector addition : uv (u1 , u 2 ) ( v 1 , v 2 ) ( u 1 v 1 , u 2 v 2 ) T(u v) T(u1 v1,u2 v2 ) ((u1 v1) (u2 v2 ),(u1 v1) 2(u2 v2 )) ((u1 u2 ) (v1 v2 ),(u1 2u2 ) (v1 2v2 )) (u1 u2 ,u1 2u2 ) (v1 v2 ,v1 2v2 ) T(u) T(v) 6.7 (2) Scalar multiplication cu c(u1,u2 ) (cu1,cu2 ) T(cu) T(cu1 ,cu2 ) (cu1 cu2 ,cu1 2cu2 ) c(u1 u2 ,u1 2u2 ) cT(u) Therefore, T is a linear transformation 6.8 Ex 3: Functions that are not linear transformations (a) f( x ) sin x sin(x x ) sin(x ) sin(x ) 1 2 1 2 (f(x) = sin x is not a linear transformation) sin( 2 3 ) sin( 2 ) sin( 3 ) (b) f( x ) x2 2 2 2 (x1 x2 ) x1 x2 (f(x) = x2 is not a linear transformation) (1 2)2 12 22 (c) f( x ) x 1 f (x1 x2 ) x1 x2 1 f (x1) f (x2 ) (x1 1) (x2 1) x1 x2 2 f (x1 x2 ) f (x1) f (x2 ) (f(x) = x+1 is not a linear transformation, although it is a linear function) In fact, f( cx ) cf ( x ) 6.9 Notes: Two uses of the term “linear”. (1) f ( x ) x 1 is called a linear function because its graph is a line (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication 6.10 Zero transformation: T :V W TV(v ) 0 , v Identity transformation: T :V V T(v) v, vV Theorem 6.1: Properties of linear transformations T :V W, u, vV (1) T(0) 0 (T(cv) = cT(v) for c=0) (2) T(v) T(v) (T(cv) = cT(v) for c=-1) (3) T(u v) T(u) T(v) (T(u+(-v))=T(u)+T(-v) and property (2)) (4) If v c1v1 c2v2 cnvn , then T(v) T(c1v1 c2v2 cnvn ) c1T(v1) c2T(v2 ) cnT(vn ) (Iteratively using T(u+v)=T(u)+T(v) and T(cv) = cT(v)) 6.11 Ex 4: Linear transformations and bases Let T : R 3 R 3 be a linear transformation such that T(1,0,0) (2,1,4) T (0,1,0) (1,5,2) T (0,0,1) (0,3,1) Find T(2, 3, -2) Sol: (2,3,2) 2(1,0,0) 3(0,1,0) 2(0,0,1) According to the fourth property on the previous slide that Tcvcv()()()()1 1 2 2 cvn n cTv 1 1 cTv 2 2 cTv n n T(2,3,2) 2T(1,0,0) 3T(0,1,0) 2T(0,0,1) 2(2,1,4) 3(1,5,2) 2T(0,3,1) (7,7,0) 6.12 Ex 5: A linear transformation defined by a matrix 3 0 2 3 v The function T : R R is defined as T(v) Av 2 1 1 v 1 2 2 (a) Find T(vv ), where (2, 1) (b) Show that TRR is a linear transformation form 23 into Sol: 2 (a) v (2, 1) R vector R3 vector 3 0 6 2 T(v) Av 2 1 3 1 1 2 0 T(2,1) (6,3,0) (b) TAAATT(u v )( u v ) u v ()() u v (vector addition) T(cu) A(cu) c(Au) cT(u) (scalar multiplication) 6.13 Theorem 6.2: The linear transformation defined by a matrix Let A be an mn matrix. The function T defined by T(v) Av is a linear transformation from Rn into Rm n m Note: R vector R vector a11 a12 a1n v1 a11v1 a12v2 a1nvn a a a v a v a v a v Av 21 22 2n 2 21 1 22 2 2n n am1 am2 amn vn am1v1 am2v2 amnvn T(v) Av T : Rn Rm ※ If T(v) can represented by Av, then T is a linear transformation ※ If the size of A is m×n, then the domain of T is Rn and the codomain of T is Rm 6.14 Ex 7: Rotation in the plane Show that the L.T. T : R 2 R 2 given by the matrix cos sin A sin cos has the property that it rotates every vector in R2 counterclockwise about the origin through the angle Sol: (Polar coordinates: for every point on the xy- v (x , y ) ( r cos , r sin ) plane, it can be represented by a set of (r, α)) r：the length of v ( xy 22 ) T(v) ：the angle from the positive v x-axis counterclockwise to the vector v 6.15 cos sin x cos sin r cos T(v) Av sin cos y sin cos r sin r cos cos r sin sin r sin cos r cos sin according to the addition formula of trigonometric r cos( ) identities r sin( ) r：remain the same, that means the length of T(v) equals the length of v +：the angle from the positive x-axis counterclockwise to the vector T(v) Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle 6.16 3 Ex 8: A projection in R The linear transformation T : R 3 R 3 is given by 1 0 0 A 0 1 0 0 0 0 is called a projection in R3 1 0 0 xx If vv is (x , y , z ), A0 1 0 y y 0 0 0 z 0 ※ In other words, T maps every vector in R3 to its orthogonal projection in the xy-plane, as shown in the right figure 6.17 Keywords in Section 6.1: function domain codomain image of v under T range of T preimage of w linear transformation linear operator zero transformation identity transformation 6.18 6.2 The Kernel and Range of a Linear Transformation Kernel of a linear transformation T : T :V W Let be a linear transformation. Then the set of all vectors v in V that satisfy T ( v ) 0 is called the kernel of T and is denoted by ker(T) ker(T) {v |T(v) 0, vV} ※ For example, V is R3, W is R3, and T is the orthogonal projection of any vector (x, y, z) onto the xy-plane, i.e. T(x, y, z) = (x, y, 0) ※ Then the kernel of T is the set consisting of (0, 0, s), where s is a real number, i.e.