Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

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1 Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89

2 9 CHAPTER 6 LINEAR TRANSFORMATION Recall, from calculus courses, a funtion f : X Y from a set X to a set Y associates to each x X a unique element f(x) Y Following is some commonly used terminologies: X is called the domain of f Y is called the codomain of f If f(x) = y, then we say y is the image of x The preimage of y is preimage(y) = {x X : f(x) = y} 4 The range of f is the set of images of elements in X In this section we deal with functions from a vector sapce V to another vector space W, that respect the vector space structures Such a function will be called a linear transformation, defined as follows Definition 6 Let V and W be two vector spaces A function T : V W is called a linear transformation of V into W, if following two prperties are true for all u,v V and scalars c T(u+v) = T(u)+T(v) (We say that T preserves additivity) T(cu) = ct(u) (We say that T preserves scalar multiplication) Reading assignment Read Textbook, Examples -, p 6- Trivial Examples: Following are two easy exampes

3 6 INTRO TO LINEAR TRANSFORMATION 9 Let V,W be two vector spaces Define T : V W as T(v) = for all v V Then T is a linear transformation, to be called the zero transformation Let V be a vector space Define T : V V as T(v) = v for all v V Then T is a linear transformation, to be called the identity transformation of V 6 Properties of linear transformations Theorem 6 Let V and W be two vector spaces Suppose T : V W is a linear transformation Then T() = T( v) = T(v) for all v V T(u v) = T(u) T(v) for all u,v V 4 If v = c v + c v + + c n v n then T(v) = T (c v + c v + + c n v n ) = c T (v )+c T (v )+ +c n T (v n ) Proof By property () of the definition 6, we have T() = T() = T() =

4 9 CHAPTER 6 LINEAR TRANSFORMATION So, () is proved Similarly, T( v) = T(( )v) = ( )T(v) = T(v) So, () is proved Then, by property () of the definition 6, we have T(u v) = T(u + ( )v) = T(u) + T(( )v) = T(u) T(v) The last equality follows from () So, () is proved To prove (4), we use induction, on n For n = : we have T(c v ) = c T(v ), by property () of the definition 6 For n =, by the two properties of definition 6, we have T(c v + c v ) = T(c v ) + T(c v ) = c T(v ) + c T(v ) So, (4) is prove for n = Now, we assume that the formula (4) is valid for n vectors and prove it for n We have T (c v + c v + + c n v n ) = T (c v + c v + + c n v n )+T (c n v n ) = (c T (v ) + c T (v ) + + c n T (v n )) + c n T(v n ) So, the proof is complete 6 Linear transformations given by matrices Theorem 6 Suppose A is a matrix of size m n Given a vector v = v v Rn define T(v) = Av = A v v v n v n Then T is a linear transformation from R n to R m

5 6 INTRO TO LINEAR TRANSFORMATION 9 Proof From properties of matrix multiplication, for u,v R n and scalar c we have and T(u + v) = A(u + v) = A(u) + A(v) = T(u) + T(v) The proof is complete T(cu) = A(cu) = cau = ct(u) Remark Most (or all) of our examples of linear transformations come from matrices, as in this theorem Reading assignment Read Textbook, Examples -, p 65-6 Projections along a vector in R n Projections in R n is a good class of examples of linear transformations We define projection along a vector Recall the definition 56 of orthogonal projection, in the context of Euclidean spaces R n Definition 64 Suppose v R n is a vector Then, for u R n define proj v (u) = v u v v Then proj v : R n R n is a linear transformation Proof This is because, for another vector w R n and a scalar c, it is easy to check proj v (u+w) = proj v (u)+proj v (w) and proj v (cu) = c (proj v (u)) The point of such projections is that any vector u R n can be written uniquely as a sum of a vector along v and another one perpendicular to v: u = proj v (u) + (u proj v (u))

6 94 CHAPTER 6 LINEAR TRANSFORMATION It is easy to check that (u proj v (u)) proj v (u) Exercise 65 (Ex 4, p 7) Let T(v,v,v ) = (v + v, v v,v v ) Compute T( 4, 5, ) Solution: T( 4, 5, ) = ( ( 4)+5, 5 ( 4), 4 ) = (,, 5) Compute the preimage of w = (4,, ) Solution: Suppose (v,v,v ) is in the preimage of (4,, ) Then (v + v, v v,v v ) = (4,, ) So, v +v = 4 v v = v v = The augmented matrix of this system is 4 its Gauss Jordan f orm So, Preimage((4,, )) = {(574, 8574, 574)}

7 6 INTRO TO LINEAR TRANSFORMATION 95 Exercise 66 (Ex p 7) Determine whether the function T : R R T(x,y) = (x,y) is linear? Solution: We have T((x,y) + (z,w)) = T(x + z,y + w) = ((x + z),y + w) (x,y) + (z,w) = T(x,y) + T(z,w) So, T does not preserve additivity So, T is not linear Alternately, you could check that T does not preserve scalar multiplication Alternately, you could check this failure(s), numerically For example, T((, ) + (, )) = T(, ) = (9, ) T(, ) + T(, ) Exercise 67 (Ex 4, p 7) Let T : R R be a linear transformation such that T(,, ) = (, 4, ), T(,, ) = (,, ), T(,, ) = (,, ) Compute T(, 4, ) Solution: We have (, 4, ) = (,, ) + 4(,, ) (,, ) So, T(, 4, ) = T(,, )+4T(,, ) T(,, ) = (, 4, )+(,, )+(,, ) = (, 5, ) Remark A linear transformation T : V V can be defined, simply by assigning values T(v i ) for any basis {v,v,,v n } of V In this case of the our problem, values were assigned for the standard basis {e,e,e } of R

8 96 CHAPTER 6 LINEAR TRANSFORMATION Exercise 68 (Ex 8, p 7) Let 4 A = Let T : R 5 R be the linear transformation T(x) = Ax Compute T(,,,, ) Solution: T(,,,, ) = 4 = 7 5 Compute preimage, under T, of (, 8) Solution: The preimage consists of the solutions of the linear system 4 x x x x 4 = 8 The augmented matrix of this system is 4 8 x 5 The Gauss-Jordan form is

9 6 INTRO TO LINEAR TRANSFORMATION 97 We use parameters x = t,x 4 = s,x 5 = u and the solotions are given by x = 5 + t + 5s + 4u,x = t,x = 4 + 5s,x 4 = s,x 5 = u So, the preimage T (, 8) = {(5+t+5s+4u, t, 4+5s, s, u) : t,s,u R} Exercise 69 (Ex 54 (edited), p 7) Let T : R R be the linear transformation such that T(, ) = (, ) and T(, ) = (, ) Compute T(, 4) Solution: We have to write (, 4) = a(, )+b(, ) Solving (, 4) = 5(, ) 5(, ) So, T(, 4) = 5T(, ) 5T(, ) = 5(, ) 5(, ) = (, 5) Compute T(, ) Solution: We have to write (, ) = a(, )+b(, ) Solving (, ) = 5(, ) 5(, ) So, T(, ) = 5T(, ) 5T(, ) = 5(, ) 5(, ) = (, ) Exercise 6 (Ex 6 (edited), p 7) Let T : R R the projection T(u) = proj v (u) where v = (,, )

10 98 CHAPTER 6 LINEAR TRANSFORMATION Find T(x,y,z) Solution: See definition 64 proj v (x,y,z) = v (x,y,z) v v = (,, ) (x,y,z) (,, ) ( x + y + z = Compute T(5,, 5) (,, ) = x + y + z (,, ), x + y + z, x + y + z Solution: We have ( x + y + z T(5,, 5) =, x + y + z, x + y + z ) ( =,, ) Compute the matrix of T ) Solution: The matrix is given by A =, because T(x,y,z) = x y z

11 6 KERNEL AND RANGE OF LINEAR TRANSFORMATION99 6 Kernel and Range of linear Transformation We will essentially, skip this section Before we do that, let us give a few definitions Definition 6 Let V,W be two vector spaces and T : V W a linear transformation Then the kernel of T, denoted by ker(t), is the set of v V such that T(v) = Notationally, ker(t) = {v V : T(v) = } It is easy to see the ker(t) is a subspace of V Recall, range of T, denoted by range(t), is given by range(t) = {v W : w = T(v) for some v V } It is easy to see the range(t) is a subspace of W We say the T is isomorphism, if T is one-to-one and onto It follows, that T is an isomorphism if ker(t) = {} and range(t) = W

12 CHAPTER 6 LINEAR TRANSFORMATION 6 Matrices for Linear Transformations Homework: Textbook, 6, Ex 5, 7,,, 7, 9,,, 5, 9,,, 5(a,b), 7(a,b), 9, 4, 45, 47; p 97 Optional Homework: Textbook, 6, Ex 57, 59; p 98 (We will not grade them) In this section, to each linear transformation we associate a matrix Linear transformations and matrices have very deep relationships In fact, study of linear transformations can be reduced to the study of matrices and conversely First, we will study this relationship for linear transformations T : R n R m ; and later study the same for linear transformations T : V W of general vector spaces In this section, we will denote the vectors in R n, as column matrices Recall, written as columns, the standard basis of R n is given by B = {e,e,,e n },,, Theorem 6 Let T : R n R m be a linear transformation Write T(e ) = a a,t(e ) = a a,,t(e n) = a n a n a m a m a mn

13 6 MATRICES FOR LINEAR TRANSFORMATIONS These columns T(e ),T(e ),T(e n ) form a m n matrix A as follows, a a a n A = a a a n a m a m a mn This matrix A has the property that T(v) = Av for all v R n This matrix A is called the standard matrix of T Proof We can write v R n as v v = v = v e + v e + + v n e n v n We have, Av = a a a n a a a n a m a m a mn v v v n = v a a a m +v a a a m + +v n = v T (e )+v T (e )+ +v n T (e n ) = T (v e + v e + + v n e n ) = T(v) The proof is complete Reading assignment: Read Textbook, Examples,; page 89-9 In our context of linear transformations, we recall the following definition of composition of functions Definition 6 Let a n a n a mn T : R n R m, T : R m R p

14 CHAPTER 6 LINEAR TRANSFORMATION be two linear transformations Define the composition T : R n R p of T with T as T(v) = T (T (v)) for v R n The composition T is denoted by T = T ot Diagramatically, R n T R m T=T ot T R p Theorem 6 Suppose T : R n R m, T : R m R p are two linear transformations Then, the composition T = T ot : R n R p is a linear transformation Suppose A is the standard matrix of T and A is the standard matrix of T Then, the standard matrix of the composition T = T ot is the product A = A A Proof For u,v R n and scalars c, we have T(u+v) = T (T (u+v)) = T (T (u)+t v)) = T (T (u))+t T (v)) = T(u)+T(v) and T(cu) = T (T (cu)) = T (ct (u)) = ct (T (u)) = ct(u) So, T preserves addition and scalar multiplication Therefore T is a linear transformation and () is proved To prove (), we have T(u) = T (T (u)) = T (A u) = A (A u) = (A A )u

15 6 MATRICES FOR LINEAR TRANSFORMATIONS Therefore T(e ) is the first column of A A, T(e ) is the second column of A A, and so on Therefore, the standard matrix of T is A A The proof is complete Reading assignment: Read Textbook, Examples ; page 9 Definition 64 Let T : R n R n, T : R n R n be two linear transformations such that for every v R n we have T (T (v)) = v and T (T (v)) = v, then we say that T is the inverse of T, and we say that T is invertible Such an inverse T of T is unique and is denoted by T (Remark Let End(R n ) denote the set of all linear transformations T : R n R n Then End(R n ) has a binary operation by composition The identity operation I : R n R n acts as the identitiy under this composition operation The definition of inverse of T above, just corresponds to the inverse under this composition operation) Theorem 65 Let T : R n R n be a linear transformations and let A be the standard matrix of T Then, the following are equivalent, T is invertible T is an isomorphism A is invertible And, if T is invertible, then the standard matrix of T is A

16 4 CHAPTER 6 LINEAR TRANSFORMATION Proof (First, recall definition 6, that T is isomorphism if T is to and onto) (Proof of () () :) Suppose T is invertible and T be the inverse Suppose T(u) = T(v) Then, u = T (T(u)) = T (T(v)) = v So, T is to Also, given u R n we have So, T is an isomorphism u = T(T (u)) So, T onto R n (Proof of () () :) So, assume T is an isomorphism Then, Ax = T(x) = T() x = So, Ax = has an unique solution Therefore A is invertible and () follows from () (Proof of () () :) Suppose A is invertible Let T (x) = A x, then T is a linear transformation and it is easily checked that T is the inverse of T So, () follows from () The proof is complete Reading assignment: Read Textbook, Examples 4; page 9 6 Nonstandard bases and general vector spaces The above discussion about (standard) matrices of linear transformations T had to be restricted to linear transformations T : R n R m This wa sbecause R n has a standard basis {e,e,,e n, } that we could use Suppose T : V W is a linear transformation between two abstract vector spaces V,W Since V and W has no standard bases, we cannot associate a matrix to T But, if we fix a basis B of V and B of W we can associate a matrix to T We do it as follows

17 6 MATRICES FOR LINEAR TRANSFORMATIONS 5 Theorem 66 Suppose T : V W is a linear transformation between two vector spaces V,W Let B = {v,v,,v n } be basis of V and B = {w,w,,w m } be basis of w We can write T(v ) = w w w m a a a m Writing similar equations for T(v ),,T(v n ), we get T(v ) T(v ) T(v n ) = w w w m a a a n a a a n a m a m a mn Then, for v = x v + x v + + x n v n, We have a a a n T(v) = w w w m a a a n a m a m a mn Write A = a ij Then, T is determined by A, with respect to bases B,B x x x n Exercise 67 (Ex 6, p 97) Let T(x,y,z) = (5x y + z, z + 4y, 5x + y)

18 6 CHAPTER 6 LINEAR TRANSFORMATION What is the standard matrix of T? Solution: We have T : R R We write vectors x R as columns x x = y instead of (x, y, z) Recall the standard basis e = We have T(e ) = z, e = 5 5, T(e ) =, e = So, the standard matrix of T is 5 A = Exercise 68 (Ex, p 97) Let of R, T(e ) = T(x,y,z) = (x + y, y z) Write down the standard matrix of T and use it to find T(,, ) Solution: In this case, T : R R With standard basis e,e,e as in exercise 67, we have T(e ) =, T(e ) =, T(e ) =

19 6 MATRICES FOR LINEAR TRANSFORMATIONS 7 So, the standard matrix of T is A = Therefore, T(,, ) = A = = 4 We will write our answer in as a row: T(,, ) = (, 4) Exercise 69 (Ex 8, p 97) Let T be the reflection in the line y = x in R So, T(x,y) = (y,x) Write down the standard matrix of T Solution: In this case, T : R R With standard basis e = (, ) T,e = (, ) T, we have T(e ) =, T(e ) = So, the standard matrix of T is A = Use the standard matrix to compute T(, 4) Solution: Of course, we know T(, 4) = (4, ) They want use to use the standard matrix to get the same answer We have 4 T(, 4) = A = = or T(, 4) = (4, ) 4 4

20 8 CHAPTER 6 LINEAR TRANSFORMATION Lemma 6 Suppose T : R R is the counterclockwise rotation by a fixed angle θ Then T(x,y) = cos θ sin θ x sin θ cos θ y Proof We can write x = r cos φ, y = r sin φ, where r = x + y, tanφ = y/x By definition T(x,y) = (r cos(φ + θ),r sin(φ + θ)) Using trig-formulas r cos(φ + θ) = r cosφcos θ r sin φ sin θ = x cos θ y sin θ and r sin(φ + θ) = r sin φ cos θ + r cos φ sin θ = y cos θ + x sin θ So, cos θ sin θ T(x,y) = sin θ cos θ The proof is complete x y Exercise 6 (Ex, p 97) Let T be the counterclockwise rotation in R by angle o Write down the standard matrix of T Solution: We use lemma 6, with θ = So, the standard matrix of T is A = cosθ sin θ cos sin = sin θ cos θ sin cos = 5 5

21 6 MATRICES FOR LINEAR TRANSFORMATIONS 9 Compute T(, ) Solution: We have T(, ) = A = 5 5 = + We write our answer as rows: T(, ) = (, + ) Exercise 6 (Ex, p 97) Let T be the projection on to the vector w = (, 5) in R : T(u) = proj w (u) Find the standard matrix Solution: See definition 64 T(x,y) = proj w (x,y) = w (x,y) (, 5) (x,y) x 5y w = (, 5) = (, 5) w (, 5) 6 ( ) x 5y 5x + 5y =, 6 6 So, with e = (, ) T,e = (, ) T we have (write/think everything as columns): T(e ) = So, the standard matrix is Compute T(, ) A =, T(e ) = Solution: We have T(, ) = A = We write our answer in row-form: T(, ) = ( 5, 65 6) =

22 CHAPTER 6 LINEAR TRANSFORMATION Exercise 6 (Ex 6, p 97) Let T(x,y,z) = (x y + z, x y,y 4z) Write down the standard matrix of T Solution: with e = (,, ) T,e = (,, ) T,e = (,, ) T we have (write/think everything as columns): T(e ) =, T(e ) = So, the standard matrix is A = Compute T(,, ) 4, T(e ) = 4 Solution: We have T(,, ) = A = 4 = 7 7 Exercise 64 (Ex 4, p 97) Let T : R R, T (x,y) = (x y, x + y) and T : R R, T (x,y) = (y, ) Compute the standard matrices of T = T ot and T = T T Solution: We solve it in three steps:

23 6 MATRICES FOR LINEAR TRANSFORMATIONS Step-: First, compute the standard matrix of T With e = (, ) T,e = (, ) T we have (write/think everything as columns): T (e ) =, T (e ) = So, the standard matrix of T is A = Step-: Now, compute the standard matrix of T With e = (, ) T,e = (, ) T we have : T (e ) =, T (e ) = So, the standard matrix of T is A = Step- By theorem 6, the standard matrix of T = T T is A A = = So, T(x,y) = (x+y, ) Similarly, the standard matrix of T = T T is A A = = So, T (x,y) = (y, y) Exercise 65 (Ex 46, p 97) Determine whether T(x,y) = (x + y,x y)

24 CHAPTER 6 LINEAR TRANSFORMATION is invertible or not Solution: Because of theorem 65, we will check whether the standard matrix of T is invertible or not With e = (, ) T,e = (, ) T we have : T(e ) =, T(e ) = So, the standard matrix of T is A = Note, deta = 4 So, T is invertible and hence T is invertible Exercise 66 (Ex 58, p 97) Determine whether T(x,y) = (x y,,x + y) Use B = {v = (, ),v = (, )} as basis of the domain R and B = {w = (,, ),w = (,, ),w = (,, )} as basis of codomain R Compute matrix of T with respect to B,B Solution: We use theorem 66 We have T(u ) = T(, ) = (,, ), T(u ) = T(, ) = (,, ) We solve the equation: (,, ) = aw + bw + cw = a(,, ) + b(,, ) + c(,, ) and we have (,, ) = (,, ) (,, )+(,, ) = w w w

25 64 TRANSITION MATRICES AND SIMILARITY Similarly, we solve (,, ) = aw + bw + cw = a(,, ) + b(,, ) + c(,, ) and we have (,, ) = (,, ) (,, ) + (,, ) = w w w So, the matrix of T with respect to the bases B,B is A = 64 Transition Matrices and Similarity We will skip this section I will just explain the section heading You know what are Transition matrices of a linear transformation T : V W They are a matrices described in theorem 66 Definition 64 Suppose A,B are two square matrices of size n n We say A,B are similar, if A = P BP for some invertible matrix P

26 4 CHAPTER 6 LINEAR TRANSFORMATION 65 Applications of Linear Trans Homework: Textbook, 65 Ex (a), (a), 5, 7, 9, 5, 7, 9, 4, 49, 5, 5, 55, 6, 65; page In this section, we discuss geometric interpretations of linear transformations represented by elementary matrices Proposition 65 Let A be a matrix and x T(x,y) = A y We will write the right hand side as a row, which is an abuse of natation If A =, then T(x,y) = A x y = ( x,y) T represents the reflection in y axis See Textbook, Example (a), p47 for the diagram If A =, then T(x,y) = A x y = (x, y) T represents the reflection in x axis See Textbook, Example (b), p47 for the diagram

27 65 APPLICATIONS OF LINEAR TRANS 5 If A =, then T(x,y) = A x y = (y,x) T represents the reflection in line y = x See Textbook, Example (c), p47 for the diagram 4 If A = k, then T(x,y) = A x y = (kx,y) T If k >, then T represents expansion in horizontal direction and < k <, then T represents contraction in horizontal direction See Textbook, Example Fig 6, p49 for diagrams 5 If A = k, then T(x,y) = A x y = (x,ky) T If k >, then T represents expansion in vertical direction and < k <, then T represents contraction in vertical direction See Textbook, Example Fig 6, p49 for diagrams 6 If A = k, then T(x,y) = A x y = (x + ky,y) T Then T represents horizontal shear (Assume k > ) The upperhalf plane are sheared to right and lower-half plane are sheared to left The points on the x axis reamain fixed See Textbook, Example, fig 64, p49 for diagrams

28 6 CHAPTER 6 LINEAR TRANSFORMATION 7 If A = k, then T(x,y) = A x y = (x,kx + y) T Then T represents vertical shear (Assume k > ) The righthalf-plane are sheared to upward and left-half-plane are sheared to downward The points on the y axis reamain fixed See Textbook, Example, fig 65, p49 for diagrams 65 Computer Graphics Linear transformations are used in computer graphics to move figures on the computer screens I am sure all kinds of linear (and nonlinear) transformations are used Here, we will only deal with rotations by an angle θ, around () x axis, () y axis and () z axis as follows: Proposition 65 Suppose θ is an angle Suppose we want to rotate the point (x, y, z) counterclockwise about z axis through an angle θ Let us denote this transformation by T and write T(x,y,z) = (x,y,z ) T Then using lemma 6, we have T(x,y,z) = x y z = cos θ sin θ x sin θ cos θ y z = x cos θ y sin θ x sin θ + y cos θ Similarly, we can write down the linear transformations corresponding to rotation around x axis and y axis We write down the transition matrices for these three matrices as follows: z The standard matrix for this transformation of counterclockwise

29 65 APPLICATIONS OF LINEAR TRANS 7 rotation by an angle θ, about x axis is cosθ sin θ sinθ cosθ The standard matrix for this transformation of counterclockwise rotation by an angle θ, about y axis is cos θ sinθ sin θ cosθ The standard matrix for this transformation of counterclockwise rotation by an angle θ, about z axis is cos θ sin θ sin θ cos θ Reading assignment: Read Textbook, Examples 4 and 5; page 4-4 Exercise 65 (Ex 6, p 44) Let T(x,y) = (x, y) (This is a vertical expansion) Sketch the image of the unit square with vertices (, ), (, ), (, ), (, ) Solution: We have T(, ) = (, ), T(, ) = (, ), T(, ) = (, ), T(, ) = (, )

30 8 CHAPTER 6 LINEAR TRANSFORMATION Diagram:,,,,,, Here, the solid arrows represent the original rectangle and the brocken arrows represent the image Exercise 654 (Ex, p 44) Let T be the reflection in the line y = x Sketch the image of the rectangle with vertices (, ), (, ), (, ), (, ) Solution: Recall, (see Proposition 65 ()) that T(x, y) = (y, x) We have T(, ) = (, ),T(, ) = (, ),T(, ) = (, ),T(, ) = (, ) Diagram:,,,,,,, Here, the solid arrows represent the original rectangle and the brocken arrows represent the image

31 65 APPLICATIONS OF LINEAR TRANS 9 Exercise 655 (Ex 8, p 44) Suppose T is the expansion and contraction represented by T(x,y) = ( x, ) y Sketch the image of the rectangle with vertices (, ), (, ), (, ), (, ) Solution: Recall, (see Proposition 65 ()) that (x, y) = (y, x)we have T(, ) = (, ),T(, ) = (, ),T(, ) = (, ),T(, ) = (, ) Diagram:,,,,, 5,,, Here, the solid arrows represent the original rectangle and thebrocken arrows represent the image Exercise 656 (Ex 44, p 44) Give the geometric description of the linear transformation defined by the elementary matrix A = Solution: By proposition 65 (6) this is a horizontal shear Here, T(x,y) = (x + y,y)

32 CHAPTER 6 LINEAR TRANSFORMATION Exercise 657 (Ex 5 and 54, p 45) Find the matrix of the transformation T that will produce a 6 o rotation about the x axis Then compute the image T(,, ) Solution: By proposition 65 () the matrix is given by A = cos θ sin θ = cos 6 o sin 6 o = sinθ cos θ So, T(,, ) = A = sin 6 o cos 6 o = + Exercise 658 (Ex 64, p 45) Determine the matrix that will produce a 45 o rotation about the y axis followed by 9 o rotation about the z axis Then also compute the image of the line segment from (,, ) to (,, ) Solution: We will do it in three (or four) steps Step- Let T be the rotation by 45 o about the y axis By proposition 65 () the matrix of T is given by A = cos θ sinθ cos 45 sin 45 = sin θ cos θ sin 45 cos 45 = Step- Let T be the rotation by 9 o rotation about the z axis By proposition 65 () the matrix of T is given by B = cos θ sin θ cos 9 o sin 9 o sin θ cos θ = sin 9 o cos 9 o =

33 65 APPLICATIONS OF LINEAR TRANS Step- So, thematrix of the composite transformation T = T T is matrix BA = = The Last Part: So, T(,, ) = BA = =

34 CHAPTER 6 LINEAR TRANSFORMATION

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