Transformations A transformation T:VW is a function that maps elements from vector space V to W The function f(x, y) = x2 + 2y is a transformation because it maps R2 into R Linear Transformations 1.Introduction to Linear Transformations 2.The Kernel and Range of a Linear Transformation 3.Matrices for Linear Transformations 1.Introduction to Linear Transformations A function T that maps a vector space V into a vector space W: mapping: , , : vector spaces T V W V W 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 = v wthen w is called the image of v under T(For each v, there is only one w) The range ofT The set of all images of vectors in V (see the figure on the next slide) 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 codomain 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 The preimage ofw 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 Ex 1: A function from R2 into R2 2 2: R R T ) 2 , ( ) , (2 1 2 1 2 1v v v v v v T + =22 1) , ( R v v e = v(a) Find the image of v=(-1,2)(b) Find the preimage of w=(-1,11) Sol: (a) ( 1,2)( ) ( 1,2) ( 1 2, 1 2(2)) ( 3,3) T T= = = + = vv(b) ( ) ( 1,11) T = = v w) 11 , 1 ( ) 2 , ( ) , (2 1 2 1 2 1 = + = v v v v v v T11 2 12 12 1= + = v vv v4 , 32 1= = v v Thus {(3, 4)} is the preimage of w=(-1, 11) Linear Transformation true are properties two following the if into of nnsformatio linear tra A :spaces vector ,W V W V TW VV T T T e + = + v u v u v u , ), ( ) ( ) ( (1)R c cT c T e = ), ( ) ( ) 2 ( u u Notes: (1) A linear transformation is said to be operation preserving ) ( ) ( ) ( v u v u T T T + = +Addition in V Addition in W ) ( ) ( u u cT c T =Scalar multiplication in V Scalar multiplication in W (2) A linear transformationfrom a vector space into itself is called a linear operator V V T :(because the same result occurs whether the operations of addition and scalar multiplication are performed before or after the linear transformation is applied) Ex 2: Verifying a linear transformation T from R2 into R2 Pf: ) 2 , ( ) , (2 1 2 1 2 1v v v v v v T + =number real any: , invector: ) , ( ), , (22 1 2 1c R v v u u = = v u1 2 1 2 1 1 2 2(1) Vector addition :( , ) ( , ) ( , ) u u v v u v u v + = + = + + u v) ( ) () 2 , ( ) 2 , ()) 2 ( ) 2 ( ), ( ) (()) ( 2 ) ( ), ( ) (() , ( ) (2 1 2 1 2 1 2 12 1 2 1 2 1 2 12 2 1 1 2 2 1 12 2 1 1v uv uT Tv v v v u u u uv v u u v v u uv u v u v u v uv u v u T T+ =+ + + =+ + + + =+ + + + + =+ + = +) , ( ) , ( tion multiplica Scalar) 2 (2 1 2 1cu cu u u c c = = u) () 2 , () 2 , ( ) , ( ) (2 1 2 12 1 2 1 2 1uucTu u u u ccu cu cu cu cu cu T c T=+ =+ = =Therefore, T is a linear transformation Ex 3: Functions that are not linear transformations (a)( ) sin f x x =2(b)( ) f x x =(c)( ) 1 f x x = +) sin( ) sin( ) sin(2 1 2 1x x x x + = +) sin( ) sin( ) sin(3 2 3 2t t t t+ = +222122 1) ( x x x x + = +2 2 22 1 ) 2 1 ( + = +1 ) (2 1 2 1+ + = + x x x x f2 ) 1 ( ) 1 ( ) ( ) (2 1 2 1 2 1+ + = + + + = + x x x x x f x f) ( ) ( ) (2 1 2 1x f x f x x f + = +(f(x) = sin x is not a linear transformation) (f(x) = x2 is not a linear transformation) (f(x) = x+1 is not a linear transformation, although it is a linear function) In fact,( ) ( ) f cx cf x = Notes: Two uses of the term linear. (1)is called a linear function because its graph is a line 1 ) ( + = x x f(2)is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication 1 ) ( + = x x f Zero transformation: V W V T e v u,, :( ) ,T V = e v 0 v Identity transformation: V V T : V T e = v v v, ) ( Theorem 1 : Properties of linear transformations W V T :0 0 = ) ( (1) T) ( ) ( (2) v v T T = ) ( ) ( ) ( (3) v u v u T T T = ) ( ) ( ) () ( ) ( then , If (4)2 2 1 12 2 1 12 2 1 1n nn nn nv T c v T c v T cv c v c v c T Tv c v c v c+ + + =+ + + =+ + + =vv(T(cv) = cT(v) for c=-1) (T(u+(-v))=T(u)+T(-v) and property (2)) (T(cv) = cT(v) for c=0) (Iteratively using T(u+v)=T(u)+T(v) and T(cv) = cT(v)) Ex 4: Linear transformations and bases Let be a linear transformation such that3 3: R R T ) 4 , 1 , 2 ( ) 0 , 0 , 1 ( = T) 2 , 5 , 1 ( ) 0 , 1 , 0 ( = T) 1 , 3 , 0 ( ) 1 , 0 , 0 ( = TSol: ) 1 , 0 , 0 ( 2 ) 0 , 1 , 0 ( 3 ) 0 , 0 , 1 ( 2 ) 2 , 3 , 2 ( + = ) 0 , 7 , 7 () 1 , 3 , 0 ( 2 ) 2 , 5 , 1 ( 3 ) 4 , 1 , 2 ( 2) 1 , 0 , 0 ( 2 ) 0 , 1 , 0 ( 3 ) 0 , 0 , 1 ( 2 ) 2 , 3 , 2 (= + = + = TT T T TFind T(2, 3, -2) 1 1 2 2 1 1 2 2According to the fourth property on the previous slide that( ) ( ) ( ) ( )n n n nT c v c v c v c T v c T v c T v| | |+ + + = + + +\ .Linear Transformations Basis vectors span vector space Know where basis goes, know where rest goes So we can do the following: Transform basis Store as columns in a matrix Use matrix to perform linear transforms Linear Transforms Example: (1,0) maps to (1,2) (0,1) maps to (2,1) Matrix is ) 2 , 2 ( ) , ( y x y x y x T + + =||.|

\|1 22 1 Ex 5: A linear transformation defined by a matrix The function is defined as 3 2: R R T ((

(((

= =212 11 20 3) (vvA T v v2 3(a) Find( ), where(2, 1)(b) Show that is a linear transformation form into TT R R= v vSol: (a)(2, 1) = v(((

=((

(((

= =036122 11 20 3) ( v v A T) 0 , 3 , 6 ( ) 1 , 2 ( = T vector2R vector3R(b)( ) ( ) ( ) ( ) T A A A T T + = + = + = + u v u v u v u v) ( ) ( ) ( ) ( u u u u cT A c c A c T = = =(vector addition) (scalar multiplication) Theorem 2 : The linear transformation defined by a matrix Let A be an mn matrix. The function T defined by v v A T = ) (is a linear transformation from Rn into Rm Note: (((((

+ + ++ + ++ + +=(((((

(((((

=n mn m mn nn nn mn m mnnv a v a v av a v a v av a v a v avvva a aa a aa a aA 2 2 1 12 2 22 1 211 2 12 1 11212 12 22 211 12 11vv v A T = ) (m nR R T : vectornR vectormR If T(v) can represented by Av, then T is a linear transformation If the size of A is mn, then the domain of T is Rn and the codomain of T is Rm Show that the L.T.given by the matrix has the property that it rotates every vector in R2 counterclockwise about the origin through the angle u Ex 7: Rotation in the plane 2 2: R R T ((

=u uu ucos sinsin cosASol: ( , ) ( cos , sin ) x y r r o o = = v(Polar coordinates: for every point on the xy-plane, it can be represented by a set of (r, )) rthe length of v () othe angle from the positive x-axis counterclockwise to the vector v2 2x y = +v T(v) ((

++=((

+=((

((

=((

((

= =) sin() cos(

sin cos cos sinsin sin cos cos sincoscos sinsin coscos sinsin cos) (o uo uo u o uo u o uoou uu uu uu urrr rr rrryxA T v v rremain the same, that means the length of T(v) equals the length of vu +othe 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 u according to the addition formula of trigonometric identities is called a projection in R3 Ex 8: A projection in R3 The linear transformationis given by 3 3: R R T (((

=0 0 00 1 00 0 1A1 0 0If is ( , , ),0 1 00 0 0 0x xx y z A y yz ((( (((= = ((( ((( v v In other words, T maps every vector in R3 to its orthogonal projection in the xy-plane, as shown in the right figure Show that T is a linear transformation Ex 9: The transpose function is a linear transformation fromMmninto Mn m ) : ( ) (m n n mTM M T A A T =Sol: n mM B Ae ,) ( ) ( ) ( ) ( B T A T B A B A B A TT T T+ = + = + = +) ( ) ( ) ( A cT cA cA cA TT T= = =Therefore, T (the transpose function) is a linear transformation from Mmn into Mnm 2. The Kernel and Range of a Linear Transformation Kernel of a linear transformation T Let be a linear transformation. Then the set of all vectors v in V that satisfyis called the kernel of T and is denoted by ker(T) W V T :0 v = ) ( T} , ) ( | { ) ker( V T T e = = v 0 v v 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.ker( ) {(0, 0, ) |is a real number} T s s = Ex 2: The kernel of the zero and identity transformations (a) If T(v) = 0 (the zero transformation ), thenW V T :V T = ) ker((b) If T(v) = v (the identity transformation), thenV V T :} { ) ker( 0 = T Ex 1: Finding the kernel of a linear transformation ) : () (3 2 2 3 = M M T A A TTSol: (((

=0 00 00 0) ker(T Ex 5: Finding the kernel of a linear transformation 13 2231 1 2( )( : )1 2 3xT A x T R Rx ( ( (= = ( ( ( x x? ) ker( = TSol: 31 2 3 1 2 3 1 2 3ker( ) {( , , ) | ( , , ) (0, 0),and ( , , ) } T x x x T x x x x x x R = = e) 0 , 0 ( ) , , (3 2 1= x x x T((

=(((

((

003 2 12 1 1321xxx(((

=(((

=(((

111321ttttxxx)} 1 , 1 , 1 span{(} number real a is | ) 1 , 1 , 1 ( { ) ker( = = t t TG.-J. E.1 1 2 0 1 0 1 01 2 3 0 0 1 1 0 (( (( Ex 6: Finding a basis for the kernel (((((

== 8 2 0 0 01 0 2 0 10 1 3 1 21 1 0 2 1

and inis where , ) ( bydefined be : Let 5 4 5AR A T R R T x x xFind a basis for ker(T) as a subspace of R5 Sol: To find ker(T) means to find all x satisfying T(x) = Ax = 0. Thus we need to form the augmented matrix first | |A 0| |G.-J. E.1 2 0 1 1 0 1 0 2 0 1 02 1 3 1 0 0 0 1 1 0 2 01 0 2 0 1 0 0 0 0 1 4 00 0 0 2 8 0 0 0 0 0 0 0A = (( (( (( (( (( 0st123452 2 12 1 21 04 0 40 1x s tx s tx s t sx tx t + (((( ((((+ (((( (((( = = = + (((( (((( (((( x{ } T B of kernel for the basis one : ) 1 , 4 , 0 , 2 , 1 ( ), 0 , 0 , 1 , 1 , 2 ( = Theorem 3: The kernel is a subspace of V The kernel of a linear transformation is a subspace of the domain V W V T :) Theorem by() ( 0 0 = T Pf: V T of subsetnonemptya is ) ker( Then . of kernel in the vectors be and LetT v u0 0 0 v u v u = + = + = + ) ( ) ( ) ( T T T0 0 u u = = = c cT c T ) ( ) (Thus, ker( ) is a subspace of (according to Theorem 4.5that a nonempty subset ofis a subspace of if it is closed under vector addition and scalar multiplication)T VV VT is a linear transformation ( ker( ), ker( ) ker( )) T T T e e + e u v u v( ker( ) ker( )) T c T e e u u Corollary to Theorem 3 : 0 xx x= = A TA T R R Tm nof space solutionthe to equal is of kernel Then the. ) ( bygiventionfransforma linearthe be : Let { }( )(a linear transformation : )ker( ) ( ) | 0, (subspace of)n mn nT A T R RT NS A A R R= = = = ex xx x x The kernel of T equals the nullspace of A (and these two are both subspaces of Rn ) So, the kernel of T is sometimes called the nullspace of T Range of a linear transformation T : ) ( range bydenoted is and of rangethe called is ins any vector of images are thatinvectorsall of setThen the n. nsformatio linear tra a be : Let T TV WW V Tw} | ) ( { ) ( range V T T e = v v For the orthogonal projection of any vector (x, y, z) onto the xy-plane, i.e. T(x, y, z) = (x, y, 0) The domain is V=R3, the codomain is W=R3, and the range is xy-plane (a subspace of the codomian R3) Since T(0, 0, s) = (0, 0, 0) = 0, the kernel of T is the set consisting of (0, 0, s), where s is a real numberW W V T of subspace a is : nnsformatio linear tra a of range The Theorem 4: The range of T is a subspace of W Pf: W T of subsetnonemptya is ) ( range have we and ), range( invectors are ) ( and ) ( Since T T T v u) ( range ) ( ) ( ) ( T T T T e + = + v u v u) ( range ) ( ) ( T c T cT e = u uThus, range( ) is a subspace of (according to Theorem 4.5that a nonempty subset ofis a subspace of if it is closed under vector addition and scalar multiplication)T WW WT is a linear transformation Range of is closed under vector addition because( ), ( ), ( ) range( )TT T T T| | |+ e\ .u v u vRange of is closed under scalar multi- plication because( ) and( ) range( )TT T c T| | |e\ .u ubecauseV + e u vbecause c V e u) Theorem by() ( 0 0 = T Notes: of subspace is ) ker( ) 1 ( V T:is a linear transformation T V W of subspace is ) ( range ) 2 ( W T(Theorem 2) (Theorem 3) 0 Kernel U V T Range T UV Corollary to Theorem 4: ) ( ) ( range i.e. , of space columnthe to equal is of range The. ) ( bygivennnsformatio linear tra the be : Let A CS T A TA T R R Tm n== x x(1)According to the definition of the range of T(x) = Ax, we know that the range of T consists of all vectors b satisfying Ax=b, which is equivalent to find all vectors b such that the system Ax=b is consistent (2) Ax=b can be rewritten as Therefore, the system Ax=b is consistent iff we can find (x1, x2,, xn) such that b is a linear combination of the column vectors of A, i.e.11 12 121 22 21 21 2nnnm m mna a aa a aA x x xa a a ((( ((( (((= + + + = ((( ((( x bThus, we can conclude that the range consists of all vectors b, which is a linear combination of the column vectors of A or said. So, the column space of the matrix A is the same as the range of T, i.e. range(T) = CS(A) ( ) CS A e b( ) CS A e b Use our example to illustrate the corollary to Theorem 3: For the orthogonal projection of any vector (x, y, z) onto the xy-plane, i.e. T(x, y, z) = (x, y, 0) According to the above analysis, we already knew that the range of T is the xy-plane, i.e. range(T)={(x, y, 0)| x and y are real numbers} T can be defined by a matrix A as follows The column space of A is as follows, which is just the xy-plane 1 0 0 1 0 00 1 0 ,such that0 1 00 0 0 0 0 0 0x xA y yz (((( ((((= = (((( (((( 11 2 3 2 1 21 0 0( ) 0 1 0 ,where,0 0 0 0xCS A x x x x x x R (((( ((((= + + = e (((( (((( Ex 7: Finding a basis for the range of a linear transformation 5 4 5Let:be defined by( ) , where is and1 2 0 1 12 1 3 1 0 1 0 2 0 10 0 0 2 8T R R T A RA = ( ( (= ( ( x x xFind a basis for the range of T Sol: Since range(T) = CS(A), finding a basis for the range of T is equivalent to fining a basis for the column space of A G.-J. E.1 2 0 1 1 10 2 0 12 1 3 1 0 01 1 0 21 0 2 0 1 00 0 1 40 0 0 2 8 00 0 0 0A B (( (( ((= = (( (( 5 4 3 2 1c c c c c5 4 3 2 1w w w w w{ }{ }1 2 4 1 2 41 2 4 1 2 4, ,and are indepdnent, so, ,can form a basis for( ) Row operations will not affect the dependency among columns, ,and are indepdnent, and thus, , is a bw w w w w wCS Bc c c c c c

{ }asis for( )That is,(1,2, 1,0),(2,1,0,0),(1,1,0,2)is a basis for the range ofCS AT