3 Geometry of Linear Transformations Please refer to the diagrams at the end of the chapter. 4.2 General Linear Transformations Example 4 1. Prove that 2 2 T : → defined by T(x,y)=(x-y,x+2y) is a linear transformation. Definition If T : V → W is a function from a vector space V into a vector space W, then T is called a linear transformation from V to W if for all vectors u and v in V and all scalars c a. T(u + v) = T(u)+T(v) b. T(cu) = cT(u)
4 2. Determine whether the following relations are linear transformations. a. 2 2 T : → ,T(x,y)=(x,3y) [L] b. 2 3 T : → ,T(x,y)=( x ,xy, y ) [NL] c. P2 P1 T : → ,T(a+bx+cx2 )=b+2cx [L] Example 5 Prove that T(x,y)=(x,1) is not a linear transformation.
5 Properties of Linear Transformations Suppose T : V → W is a linear transformation, and u, v V . Then, a. T(0) = 0 b. T(- v) = −T(v) c. T(u − v) = T(u)−T(v) d. 1 2 2 n n If c c c , then = + + + 1 v v v v ( ) ( 1 2 2 n n ) 1 2 2 n n T T c c c c T( ) c T( ) c T( ) = + + + = + + + 1 1 v v v v v v v Example 6 Suppose 3 3 T : → is a linear transformation and T(1,0,0)=(2,-1,4), T(0,1,0)=(1,5,-2), T(0,0,1)=(0,3,1). Find a formula for T(a,b,c) and thus find T(2,3,-2). [T(a,b,c) = (2a+b,-a+5b+3c,4a-2b+c); (7,7,0)]
6 Example 7 Suppose 3 2 T : → is a linear transformation and S={u=(1,1,1), v=(1,1,0), w=(1,0,0)} is a basis for 3 . If T(u)=(1,0),T(v)=(2,-1), and T(w)=(4,3), find a formula for T(a,b,c) and thus find T(2,-3,5). [T(a,b,c) = (4a-2b-c,3a-4b+c); (9,23)]
7 4.3 Kernel and Range Definition (Kernel) Let T : V → W be a linear transformation. The kernel of T, denoted by ker(T) is the subset of V consisting of all vectors, v such that: T(v)=0 i.e. ker(T)={v, v V , T(v)=0} Example 8 Let 3 2 T : → be defined by T(x,y,z)=(x,y). T(0,0,2)=(0,0) (0,0,2)ker(T) T(2,-3,4)=(2,-3) (0,0) (2, = 3,4)ker(T) To find ker(T), set T(v)=0 where 3 v T(x,y,z)=(0,0) (x,y)=(0,0) i.e. x=y=0 and z is any real number. i.e. ker(T)={(0,0,t); t } and this is actually the z-axis. Example 9 If 2 2 T : → is defined by T(x,y)=(x+y,x-y), find ker(T).
8 Example 10 4 2 T : → is defined by: + + = z w x y w z y x T . Find ker(T).
9 Example 11 3 2 T : → is defined by T(x) = Ax where − − − = 1 2 3 1 1 2 A . Find ker(T), a basis for ker(T) and the dimension of ker(T). [If n m T : → is a linear transformation with standard matrix, A then, ker(T) is the null space of A.]
10 Example 12 5 4 T : → is defined by T(x) = Ax where − − − = 0 0 0 2 8 1 0 2 0 1 2 1 3 1 0 1 2 0 1 1 A . Find a basis for ker(T).
11 Example 13 T : V → W is a linear transformation. Prove that ker(T) is a subspace of V.
12 Definition (Range) If T : V → W is a linear transformation then the set of all vectors in W that are images under T of at least one vector in V is called the range of T and denoted R(T). Theorem The range of the linear transformation T : V → W is a subspace of W. Theorem Suppose n m T : → is a linear transformation defined by T(x) = Ax . Then, the range of T is the column space of A. Example 14 3 3 T : → is the orthogonal projection onto the xy-plane. Determine the kernel and range of T.
13 Example 15 Refer to Example 12 above and find a basis for the range of T.
14 Definition If T : V → W is a linear transformation, then the dimension of the range of T is called the rank of T and the dimension of the kernel of T is called the nullity of T. Example 16 6 4 T : → be a linear transformation with the following standard matrix − − − − − − − = 4 9 2 4 4 7 2 5 2 4 6 1 3 7 2 0 1 4 1 2 0 4 5 3 A . Find the rank and nullity of T. (Refer to Section 3.9 for the answer.)
15 Dimension Theorem for Linear Transformation If T : V → W is a linear transformation and dim(V)=n, then, Example 17 Find the rank and nullity of the linear transformation 3 3 T : → with the standard matrix: − = 0 0 0 0 1 1 1 0 2 A . Rank(T) + Nullity(T) = n
16 Example 18 Suppose 5 7 T : → is a linear transformation. a. Find the dimension of ker(T) if the dimension of R(T) is 2. b. Find the rank of T if nullity(T) = 4. c. Find the rank of T if ker(T) = {0}.
17 4.4 One-to-One Linear Transformations Definition The linear transformation T : V → W is said to be one-to-one if every vector w W has only one pre-image in V such that T(v)=w. In other words, T maps distinct vectors in V onto distinct vectors in W. Theorem If T : V → W is a linear transformation, then T is one-to-one if and only if ker(T) ={0}. Example 19 State whether the following linear transformation are one-to-one. a. 3 3 T : → is the orthogonal projection onto the xy-plane. b. 6 4 T : → is the transformation defined in Example 16.
18 4.5 Onto Linear Transformations Definition T : V → W is said to be onto if every vector in W has a pre-image in V i.e. the range of T=W. Theorem Let T : V → W be a linear transformation and dim(V)=n. Then, T is onto if and only if rank(T) =n. Theorem Let T : V → W be a linear transformation and dim(V)=n. Then, T is one-to-one if and only if T is onto. Example 20 The following transformations n m T : → are defined by T(x)=Ax. Determine the nullity and rank of T and hence, state whether T is one-to-one and/or onto. a. = 0 0 1 0 1 1 1 2 0 A b. = 0 0 0 1 1 2 A c. = 0 1 1 1 2 0 A d. = 0 0 0 0 1 1 1 2 0 A
19 Definition: (Isomorphism) The linear transformation T : V → W is said to be an isomorphism if T is one-to-one and onto.V and W are then said to be isomorphic to one another. 4.6 Standard Basis and the Standard Matrix If n m T : → is a linear transformation, and e1, e2, …….,en are the standard basis vectors for n then the standard matrix for T is: A = (T(e1)|T(e2)|…….|T(en)) Example 21 Suppose 3 3 T : → is the orthogonal projection on the xy-plane. Find the standard matrix for T using the standard basis vectors for 3 .
20 Example 22 2 2 T : → is the orthogonal projection onto the x-axis followed by reflection about the y-axis. Find the standard matrix of T using the standard basis vectors for 2 .
21 4.7 Compositions of Linear Transformations Definition If T1 : U →V and T2 : V → W are linear transformations, the composition of T2 with T1 denoted by T2 T1 is defined by: (x) ( (x)) 2 1 T2 T1 (T T ) = where xU T2 T1 U V W Example 23 Let 1 P1 P2 T : → and 2 P2 P2 T : → be linear transformations given by: T (p(x)) xp(x) 1 = T (p(x)) p(2x 4) 2 = + Find a formula for 2 1 P1 P2 (T T ): → . x T1(x) T2(T1(x))
22 Let n m 1 T : → and m k 2 T : → both be linear transformations with standard matrices A1 and A2 respectively. The composition n k T : → written as (x) ( (x)) 2 1 T2 T1 (T T ) = is also a linear transformation and the standard matrix for T is: A=A2A1 Example 24 The linear transformations, T1 and T2 are defined as follows: T1(x,y,z)=(2x+y,0,x+z) T2(x,y,z)=(x-y,z,y) i. Find a formula for ( T2 T1 )(x,y,z). ii. Verify that the standard matrix for T2 T1 is the same as that obtained by multiplying the standard matrices for T1 and T2. iii. Find the standard matrix for T1 T2 . iv. Is T1 T2 = T2 T1 ?
23 4.8 Inverse Transformations Let T : V → W be a linear transformation. If T is one-to-one, then each vector v in V has a unique image w=T(v) in the range of T. The inverse of T, denoted by 1 T − maps w back into v. If the standard matrix for T is A then the standard matrix for 1 T − is 1 A − . Theorem Let n n T : → be a linear transformation with standard matrix A. Then, the following statements are equivalent: 1. T is invertible. 2. T is one-to-one and onto. 3. A is invertible. Example 25 The linear transformation 3 3 T : → is defined by: T(x,y,z)=(2x+3y+z,3x+3y+z,2x+4y+z). Show that T is invertible and hence, find T ( ). 1 v −
24 Inverse of Composite Transformations If T1 : U →V and T2 : V → W are both one-to-one linear transformations, then a. T2 T1 is also one-to-one. b. 1 2 1 (T T ) − = 1 2 1 T1 T − − Example 26 Let 2 2 1 T : → and 2 2 2 T : → be the linear operations given by the formulas T (x, y) = (x + y, x − y) 1 and T (x, y) (2x y, x 2y) 2 = + − a. Show that T1 and T2 are one to one. b. Find formulas for T (x, y) -1 1 ,T (x, y) -1 2 and (T T ) (x, y) 1 2 1 − . c. Verify that ( ) -1 2 -1 1 1 T2 T1 = T T − .
25 GEOMETRY OF LINEAR TRANSFORMATIONS Reflection Operators in R2 Operator Illustration Equations Standard Matrix Reflection about the y-axis Reflection about the x-axis Reflection about the line y=x Reflection Operators in R3 Operator Illustration Equations Standard Matrix Reflection about the xyplane Reflection about the xzplane Reflection about the line yzplane
26 Projection Operators in R2 Operator Illustration Equations Standard Matrix Orthogonal projection on the x-axis Orthogonal projection on the y-axis Projection Operators in R3 Operator Illustration Equations Standard Matrix Orthogonal projection on the xy-plane Orthogonal projection on the xz-plane Orthogonal projection on the yz-plane
27 Contraction & Dilation Operators in R2 Operator Illustration Equations Standard Matrix Contraction with factor k on R2 (0≤k≤1) Dilation with factor k on R2 (k≥1)
Exercise 4 1. Find the standard matrix for each of the following linear transformations: a. T(x,y)=(x+y,2x-3y) b. T(x,y)=(x+y,9y-4x,-x+2y,3x) c. T(x,y)=(x-6y,x+5y,y-8x) d. T(x,y)=(5x+y,0,2x+8y) e. T(x,y,z)=(-x+y,x-4y,2z) f. T(x,y,z)=(5x-3y+z,2z+4y,5x+3y) g. T(x,y,z)=(z-y,5x+18z) h. T(x1,x2,x3,x4)=(0,0,0) 2. Use the standard matrix for the linear transformation, T to find the image of v: a. T(x1,x2,x3,x4)=(x1+x2,-x2+x3+2x4); v=(-1,1,-1,1) b. T(x1,x2,x3,x4) =(3x1-2x3,x1+x2+x3,4x3+x2,x2+x4); v=(-1,3,-3,-2) c. T(x,y,z)=(3x+7y-z,4x+4y+4z); v=(-1,2,3) d. T(x,y)=(x+y,x-2y,x,-y); v=(4,2) 3. Determine whether the following transformations 2 2 T : → are linear. a. T(x,y)=(x,x+y) b. T(x,y)=(x,y2 ) c. T(x,y)=(-y,x) d. T(x,y)=(0,y) e. T(x,y)=(x-2y,2x+y) f. T(x,y)=(2x,y+1) g. T(x,y)=(x,x) h. T(x,y)=( x, y ) 4. Determine whether the following transformations 3 2 T : → are linear. a. T(x,y,z)=(x+y+z,y) b. T(x,y,z)=(2,2) c. T(x,y,z)=(0,0) d. T(x,y,z)=(2x+3y,-5x+4y) 5. Determine whether the following transformations are linear. a. T : M22 → where a 2b c - d c d a b T = + + b. T : M22 → where 2 2 a b c d a b T = + c. 22 T : 2 → M where = 0 xy x y y x T d. P2 P2 T : → where ( ) 2 0 1 2 2 T a 0 + a1x + a 2 x = a + a (x +1) + a (x +1) e. P2 P2 T : → where ( ) 2 0 1 2 2 T a 0 + a1x + a 2 x = (a +1) + (a +1)x + (a +1)x f. P1 P1 T : → where T(a + bx) = (2a + b) + (a - 3b)x g. P1 P1 T : → where T(a + bx) = (a − b) + (a + b +1)x h. P2 P1 T : → where T(a bx cx ) (a - b) (b c)x 2 + + = + + i. 2 M22 T : → where = c a c d a b T j. 22 T : 2 → M where = 0 2 a b b a T k. → 3 T : where T(x) = x • x
l. 3 M22 T : → where (a,b, c d) c d a b T = + m. 3 M22 T : → where (x y, z,w 1) z w x y T = + + 6. The transformation 22 T : 2 → M is defined as follows: ( ) = 0 2x x - y 0 T x, y . a. Determine whether T(u+kv) = T(u) +T(kv) for any u,v in 2 . b. What can be concluded from (a)? 7. Suppose T : V → V is a linear transformation. If T(u+v) = u-v and T(u-2v) = -2u+v,find: a. T(u) and T(v) in terms of u and v. b. T(3u-2v) in terms of u and v. 8. Consider the transformation M2x3 P2 T : → such that 2 ab cx fx d e f a b c T = + + . a. What are the domain and codomain of T? State their dimensions. b. Determine whether T is linear. 9. Find T(a,b,c) if → 3 T : is a linear transformation such that: T(1,1,1)=3; T(0,1,-2)=1; T(0,0,1)=-2 Hence, find the image of the vector (-1,-2,4). 10. Consider the basis S={v1=(-2,1), v2=(1,3)} for 2 ,and let 2 3 T : → be a linear transformation such that T(v1)=(-1,2,0) and T(v2)=(0,-3,5). Find a formula for T(x,y) and use that formula to find T(2,-3). 11. Consider the basis S={v1=(1,1), v2=(1,0)} for 2 ,and let 2 2 T : → be a linear transformation such that T(v1)=(1,-2) and T(v2)=(-4,1). Find a formula for T(x,y) and use that formula to find T(2,-3). 12. Consider the basis S={v1=(1,1,1), v2=(1,1,0), v3=(1,0,0)} for 3 ,and let 3 3 T : → be a linear transformation such that T(v1)=(2,-1,4), T(v2)=(3,0,1) and T(v3)=(-1,5,1) . Find a formula for T(x,y,z) and use that formula to find T(2,4,-1). 13. Consider the basis S={v1=(1,2,1), v2=(2,9,0), v3=(3,3,4)} for 3 ,and let 3 2 T : → be a linear transformation such that T(v1)=(1,0), T(v2)=(-1,1) and T(v3)=(0,1). Express (7,13,7) as a linear combination of v1, v2 and v3 and hence find T(7,13,7). 14. Let T be the linear transformation such that T(1,-1)=(5,-8) and T(2,2)=(-9,4). Find T(2,-3). 15. Let P1 P2 T : → be the linear transformation such that T(-1+x)=-8+5x and
T(2+2x)=4-9x. Find T(a+bx) and hence, find T(-15+10x). 16. Let P2 P1 T : → be the linear transformation such that T(1+x+x2 )=-1+3x, T(1+x-x 2 )=-3+2x and T(1-x+x2 )=1+2x. Find T(a+bx+cx2 ) and hence, find T(-25+15x-10x2 ). 17. 2 M22 T : → is a linear transformation such that: (3,5) 0 3 2 0 T = and (-1,2) 2 0 0 5 T = − . By expressing 4 9 6 -10 as a linear combination of 0 3 2 0 and - 2 0 0 5 , find T 4 9 6 -10 . 18. Let 2 2 T : → be the linear operator defined by: T(x,y)=(2x-y,-8x+4y). Which of the following vectors is in the kernel of T? a. (5,10) b. (3,2) c. (1,1) 19. Let 4 3 T : → be the linear transformation defined by: T(x1,x2,x3,x4)=(4x1+x2-2x3-3x4, 2x1+x2+x3-4x4, 6x1-9x3+9x4) Which of the following vectors is in ker(T)? a. (3,-8,2,0) b. (0,0,0,1) c. (0,-4,1,0) 20. Find the standard matrices for T T1 T2 = and 2 1 ' T = T T and state whether T1 T2 T2 T1 = . a. T : , T (x, y) (x - 2y,2x 3y) T : , T (x, y) (2x, x y) 2 2 2 1 2 2 2 1 → = + → = − b. T : , T (x, y, z) (x, y, z) T : , T (x, y, z) (0, x,0) 2 3 3 1 2 3 3 1 → = → = c. T : , T (x, y) (x, y, y) T : , T (x, y, z) (y, z) 2 3 2 1 2 2 3 1 → = → = 21. Find ker(T), and state whether T is one-to-one. a. T : , where T(x, y) (y, x) 2 2 → = b. T : , whereT(x, y) (0,2x 3y) 2 2 → = + c. T : , where T(x, y) (x y, x - y) 2 2 → = + d. T : , whereT(x, y) (x, y, x y) 2 3 → = + e. T : , whereT(x, y) (x - y, y - x,2x - 2y) 2 3 → = f. T : , whereT(x, y, z) (x y z, x - y - z) 3 2 → = + + 22. Show that the following linear transformations are onto. a. T : , whereT(x, y) (x - y,-x 2y) 2 2 → = + b. T : , whereT(x, y, z) (x, z) 3 2 → = c. T : , where T(x, y, z) (x y z,-x y z) 3 2 → = − + + + d. T : P P , where T(a bx cx ) (a b) (a c)x 2 2 → 1 + + = + + + e. 2 2 2 2 T : P → P , where T(a + bx + cx ) = c + bx + ax
23. For the given linear transformations n m T : → find (i) n and m (ii) kernel (T) (iii) the range of T. a. T(x, y) = (x + y, x - y) b. T(x, y) = (x - y,0) c. T(x, y, z) = (x + y, x - z) d. T(x, y) = (x - y, x − y,0) e. T(x, y, z) = (x - z,-x + z, x - z) f. − = y x 1 0 1 2 y x T g. (v) v − = 0 1 1 0 1 0 1 2 T 24. T is the multiplication by the given matrix A. Find (i) a basis for ker(T) (ii) a basis for the range of T (iii) the rank and nullity of T (iv) the rank and nullity of A. a. − − = 7 4 2 5 6 4 1 1 3 A b. − − = 0 0 0 4 0 2 2 0 1 A 25. Let 2 2 T : → be the multiplication by the given matrix A. Determine whether T is invertible. If so, find T (x, y) −1 . a. = 2 1 5 2 A b. − − = 4 2 6 3 A c. − = 1 3 4 7 A 26. Let 3 3 T : → be the multiplication by the given matrix A. Determine whether T is invertible. If so, find ( ) 1 2 3 1 T x , x , x − . a. − = 1 1 0 1 2 1 1 5 2 A b. − − = 2 3 0 0 2 1 1 1 1 A 27. Determine whether multiplication by the given matrix A is a one-to-one linear transformation. a. − − − = 3 6 2 4 1 2 A b. − = 5 3 1 5 4 2 A 28. 4 3 T : → is a linear transformation defined by: + + + = + x 2y z w y 2z x - 3z w z y x T . a. Find the standard matrix A that represents T.
b. Find kernel (T) and nullity (T). c. Is T a one-to-one linear transformation? Give a reason for your answer. d. Find a basis for the range of T and state the rank of T. e. Is T onto? Give a reason for your answer. 29. P1 P1 T : → is a linear transformation such that T(1-x)= 2+5x and T(2)= 2x. Find T (3(2 5x) 5(2x)). -1 + − 30. P2 P1 T : → is the linear transformation defined by: T(a+bx+cx2 )= (a+b)+(-a+b+c)x. Find: a. A basis for the kernel of T. b. A basis for the range of T. 31. Consider the set B={p=x+1, q=x+x2 , r=1+x2 } a. Show that B is linearly independent. b. Express 2+4x-6x2 as a linear combination of p,q and r. c. P2 P2 T : → is a linear transformation where P2 refers to the set of polynomials of degree 2 and T(x+1)=x2 -1 T(x+x2 )=x2 -2x T(1+x2 )=-x 2+3x Find T(2+4x-6x2 ). 32. a. Suppose T is a transformation. i. What are the conditions that T must satisfy so that it is a linear transformation? ii. State the conditions that T must satisfy so that it is onto. iii. If ker(T)={0}, find nullity (T) and rank (T). iv. What conditions would make T invertible. b. 4 3 T : → is a linear transformation defined by: + + + + + + + = 2y 2z 2w 3x 2y 5z 5w x z w w z y x T . i. What is the standard matrix, A for the above linear transformation? ii. Find ker(T). iii. Is T one-to-one? Give a reason for your answer. c. P2 P3 T : → is the linear transformation defined by: T(ax2+bx+c)= ax3+bx2+cx. P2 refers to the set of polynomials of degree 2 and P3 refers to the set of polynomials of degree 3; a,b and c are constants. i. Determine whether T(ku+v)=kT(u)+T(v). ii. What can you conclude about T from (i)? 33. Consider the transformation 2 3 T : → that is defined by: T(x,y)= (x+y,-x+y,2x+y). a. Obtain the standard matrix for the above transformation. b. What is meant by ker(T)? Find ker(T). c. Find a basis for the range of T. d. State the rank and nullity of T.
e. Is T one-to-one? Is T onto? Give reasons for your answer. 34. Consider the vectors u, v and w where u=(1,2,0), v=(0,1,-1) and w=(1,0,3). Let 2 3 T : → P be a transformation such that: T(u)=1+x, T(v)=1-x 2 and T(w)=x+x2 a. Express any vector (a,b,c) in 3 as a linear combination of u,v and w. b. Show that T(a,b,c)=(-3a+2b+c)+ax+(4a-2b-c)x2 . c. Find T(1,0,-1). d. Find a basis for the kernel of T. e. Find a basis for the range of T. 35. Consider the following linear transformation 3 2 T : → where: T(x,y,z)=(x+y,2x-z) a. State the domain of T and the codomain of T. b. Find the kernel of T. c. Is T one-to-one? Explain your answer. d. Is T onto? Explain your answer. 36. Suppose 2 4 T : → P is a linear transformation that is defined by: T(a,b,c,d)=(a+2d)+(b-c)x+(a-b+c+d)x2 a. Find a basis for the kernel of T. b. Find a basis for the range of T. c. Is T one-to-one transformation? Give a reason for your answer. d. Is T onto? Give a reason for your answer. 37. Consider the linear transformation 5 4 T : → given by T(x)=Ax where = 2 3 5 1 8 -1 0 -1 0 -1 3 - 2 1 0 -1 1 4 5 0 9 A a. Find a basis for the kernel of T. What is the nullity of T? b. Find a basis for the range of T. What is the rank of T? c. Explain whether T one-to-one and onto. 38. Consider a linear transformation 3 3 T : → defined as follows: + − + = - y 3z 3x - y x 2y - z z y x T a. Find the standard matrix, A for T. b. Find the kernel of T. c. Is T one-to-one? Give a reason for your answer. d. Find dim(domain), dim(kernel) and dim(range). Is T onto? 39. Consider the linear transformation 5 4 T : → that is defined by:
+ + + + + + + + + + + + + = 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 5 4 3 2 1 3x - 4x x 2x 3x x - x 3x 5x 6x - x 2x 3x 4x 5x x - x 2x 3x 4x x x x x x T a. Obtain the standard matrix, A of the transformation, T. b. Find a basis for the kernel of T and state the nullity of T. c. Find a basis for the range of T and state the rank of T. d. Is T a one-to-one transformation? Give a reason for your answer. e. Is T onto? Give a reason for your answer. 40. Suppose 3 3 T : → is a linear transformation defined as follows: T(a,b,c)=(a+b+c, 2a+b, 3b+2c) a. Find the standard matrix of T. b. Find the kernel and nullity of T. c. Is T one-to-one? Give a reason for your answer.
Answer 2. a. (0,0) b. (3, 1, 9,1 − − ) c. (8,16) d. (6,0,4, 2− ) 4. a. Linear b. Not linear c. Linear d. Linear 5. a. Linear b. Not linear c. Not linear d. Linear e. Not linear f. Linear g. Not linear h. Linear i. Linear j. Not linear k. Not linear l. Linear m. Not linear 6. a. Yes b. Linear 8. a. Domain of T = M2x3 Dimension = 6; Codomain of T = P2 Dimension = 3 b. Not linear 10. ( ) 3 9 4 5 10 T , , , 7 7 7 x y x y x y x y − − − + = ; ( ) 9 6 20 T 2, 3 , , 7 7 7 − = − − 12. T , , 4 , 5 5 ,3 ( x y z y z x z x y z x ) = − − − + − + ( ) ; T 2,4, 1 15, 9, 1 ( − = − − ) ( ) 14. ( ) 59 T 2, 3 , 21 4 − = − 16. ( ) ( ) 2 4 T 2 abc a bx cx a b c x + + + + = − − + + ; ( ) 2 95 T 25 15 10 2 − + − = − x x x 18. a. kernel b. not kernel c. not kernel 20. a. Standard matrix 1 2 0 2 T T 7 3 = − ; Standard matrix 2 1 2 4 T T 1 5 − = − − ; T T T T 1 2 2 1 b. Standard matrix 1 2 000 T T 1 0 0 000 = ; Standard matrix 2 1 000 T T 1 0 0 000 = ; T T =T T 1 2 2 1 c. Standard matrix 1 2 0 1 0 T T 0 0 1 0 0 1 = ; Standard matrix 2 1 0 1 T T 0 1 = ; T T T T 1 2 2 1
24. a.i) 14 11 19 11 1 − ii) 1 1 5 , 6 7 4 − iii) 2,1 iv) 2,1 b.i) 1 2 0 0 , 1 1 0 − ii) 2 4 0 iii) 1,2 iv) 1,2 26. a. T is not invertible b. T is invertible. ( ) ( ) 1 T x ,x ,x 3x +3x -x , 2x -2x +x , 4x -5x +2x , 1 2 3 1 2 3 1 2 3 1 2 3 − = − − 28. a. 1 0 3 0 0 1 2 0 1 2 1 1 − b. 3 2 , 1 0 m m − ,1 c. No because Nullity (T) = 1 0 d. 1 0 0 0 , 1 , 0 1 2 1 ,3 e. Yes because Rank (T) = 3 = Dim ( ) 3 . 30. a. 1 1 2 2 2 x x − + b. 1 ,1 − + x x 32. a. i) If T : V → W is a function from a vector space V into a vector space W, then T is called a linear transformation from V to W if for all vectors u and v in V and all scalars c a. T(u + v) = T(u)+ T(v) b. T(cu) = cT(u) ii) Let T : V → W be a linear transformation and dim(V)=n. Then, T is onto if and only if rank(T) =n.
iii) Nullity (T) = 0,Rank (T) = no. of columns iv) Standard matrix, A is invertible OR T is one to one and onto. b. i) Standard matrix, A 1 0 1 1 3 2 5 5 0222 ii) 1 1 1 1 , , 1 0 0 1 m n m n − − − − + iii) No because Nullity (T) = 2 0 c. i) Yes ii) T is linear. 34. a. (a b c a b c b a c c b a , , 3 3 6 2 2 ) = − − + − + + + − ( )u v w ( ) ( ) c. 2 − + + 4 5 x x d. 0 1 2 1 − e. 2 2 − + + − 3 4 ,2 2 x x x 36. a. 0 1 1 0 b. 222 1 , ,2 + − + x x x x c. No because Nullity (T) = 1 0 d. Yes because Rank (T) = 3 = Dim (P2 ) . 38. a. Standard matrix, A for T 1 2 1 3 1 0 0 1 3 − − − − b. 0 c. Yes because Nullity (T) = 0 d. 3,0,3,Yes
40. a. Standard matrix of T 111 2 1 0 0 3 2 b. 0 ,0 c. Yes because Nullity (T) = 0
1 CHAPTER 5 EIGENVALUES & EIGENVECTORS 5.1 Eigenvalues and Eigenvectors Definition Let A be anxn matrix. The real number is called an eigenvalue of A if there exists a non-zero vector x in n such that x is called an eigenvector of A corresponding to . Note that x = 0 satisfies Ax = x but 0 is NOT an eigenvector of A by definition. Geometric Interpretation in 2 x x x x x x x x 1 0 1 −1 −1 0 The linear operator Ax = x compresses or stretches x by a factor of . Example 1 The vector = 2 1 x is an eigenvector of = 8 -1 3 0 A corresponding to the eigenvalue of = 3. Since: x = x = = − = 2 1 3 6 3 2 1 8 1 3 0 A Ax= x
2 Computing Eigenvalues and Eigenvectors In general, to find eigenvalues of A, we write: ( )x 0 x x x x = = = I - A A I A In order to obtain non-trivial solutions to this system, we require I - A = 0 This equation is called the characteristic equation of A. When expanded, I - A is a polynomial in called the characteristic polynomial of A. Example 2 Find the eigenvalues of = - 2 4 1 1 A .
3 Example 3 Find the eigenvalues of 1 0 3 A 1 1 2 1 1 2 = − − − . Example 4 Find the eigenvalues of the upper triangular matrix = 0 0 f 0 d e a b c A .
4 Theorem If A is anxn triangular (upper, lower, diagonal) matrix, then the eigenvalues of A are the entries on the main diagonal. Finding Eigenvectors and Bases for Eigenspaces Eigenvectors are the non-zero vectors that satisfy Ax = x i.e. they are non-zero vectors in the solution space of (I- A)x = 0 . This solution space is called the eigenspace of A corresponding to . Example 5 Find bases for the eigenspaces of = 1 0 3 1 2 1 0 0 - 2 A .
5 Example 6 Find bases for the eigenspaces of − = 1 1 - 2 1 -1 2 1 0 3 A .
6 Eigenvalues of Ak Let be the eigenvalue of A and x be the corresponding eigenvector [i.e.Ax = x] A 2x = A(Ax)=A( x)= (Ax)= ( x)= 2 x i.e. 2 is an eigenvalue of A2and x is a corresponding eigenvector. Theorem If k is a positive integer, is an eigenvalue of matrix A and x a corresponding eigenvector, then k is an eigenvalue of Ak and x is a corresponding eigenvector. Theorem A square matrix, A is invertible if and only if = 0 is NOT an eigenvalue of A.
7 Eigenvalues, Eigenvectors and Linear Transformations Definition Let n n T : → be a linear operator. A scalar is called an eigenvalue of T if there is a non-zero vector n x such that T(x) = x x is called the eigenvector of T corresponding to . Observe that if A is the standard matrix for T then, T(x)=Ax = x i.e. • The eigenvalues of T are the eigenvalues of its standard matrix, A. • x is an eigenvector of T corresponding to if and only if x is an eigenvector of A corresponding to . Example 7 Let 3 3 T : → be the orthogonal projection on the xy-plane. a. Find the eigenvalues of T. b. Find the eigenvectors of T.
8 Exercise Consider the following linear operator: + + = - y 2z 3y - 2z 2x - 2y 3z z y x T a. What is the matrix representation, A of T? b. Find the eigenvalues of T. c. Find the eigenvectors and the bases for the eigenspaces corresponding each eigenvalue. d. Find the kernel of T and the nullity of T. e. State whether T is one-to-one. If so, find z y x T -1 .
9 5.2 Diagonalisation Definition A square matrix A is said to be diagonalisable if there exists an invertible matrix P such that P -1AP is a diagonal matrix. Matrix P is said to diagonalise A. Theorem If A is anxn matrix then the following are equivalent: 1. A is diagonalizable. 2. A has n linearly independent eigenvectors. Procedure for Diagonalising a Matrix 1. Find n linearly independent eigenvectors of A say, p1,p2,…….,pn. 2. Form the matrix P having p1,p2,…….,pn as its column vectors. 3. The matrix P -1AP will then be a diagonal matrix with 1 2 n , , , as its successive diagonal entries, where is the eigenvalue corresponding to pi, for i=1,2,…..,n.
10 Example 8 Find a matrix P that diagonalises − = 1 0 3 1 2 1 0 0 2 A From the example 5.1 we obtained the following eigenvalues and eigenvectors: = = = 0 1 0 1 0 -1 2 p1 p2 = = 1 1 - 2 1 p3 Since there are 3 basis vectors, matrix A is diagonalisable. − − = 1 0 1 0 1 1 1 0 2 P − − = 1 0 1 1 1 1 1 0 2 P -1 − − − − − = 1 0 1 0 1 1 1 0 2 1 0 3 1 2 1 0 0 2 1 0 1 1 1 1 1 0 2 P AP -1 = 0 0 1 0 2 0 2 0 0
11 If the columns of P are rearranged, then the diagonal entries of P -1APwill be rearranged accordingly. If − − = 1 1 0 0 1 1 1 2 0 P then = 0 0 2 0 1 0 2 0 0 P AP -1 Example 9 Find a matrix P that diagonalises = - 3 5 2 1 2 0 1 0 0 A .
12 Theorem If an nxn matrix A has n distinct eigenvalues, then A is diagonalisable. Theorem A is diagonalisable if and only if for each eigenvalue, ,the geometric and algebraic multiplicities of are equal.
13 Computing Powers of a Matrix If A is a nxn matrix and P is an invertible matrix, then ( ) P A P P AIAP P AP P APP AP -1 2 -1 -1 -1 2 -1 = = = In general, ( ) k k k -1 k -1 k -1 P A P = P AP = D A = PD P Since D is a diagonal matrix, Dk is easy to compute. Recall that if D= n 2 1 0 0 d 0 d 0 d 0 0 then Dk= k n k 2 k 1 0 0 d 0 d 0 d 0 0 Example 10 Find A13 if = 1 0 3 1 2 1 0 0 - 2 A .
1 Exercise 5 1. Verify that i is an eigenvalue of A and that xi is a corresponding eigenvector. a. ( ) ( ) 3, (5,1,2) 1, 1, 1,0 2, 1,0,0 0 0 3 0 -1 2 2 3 1 A 3 2 2 1 = = = − = − = = = 3 1 x x x b. ( ) ( ) 3, (3,0,1) 3, 2,1,0 5, 1,2, 1 -1 - 2 0 2 1 - 6 - 2 2 - 3 A 3 2 2 1 = − = = − = − = = − = 3 1 x x x 2. For each of the following matrices, find (i) the characteristic equation (ii) the eigenvalues and (iii) the bases of the corresponding eigenspaces. a. − − 2 1 6 3 b. 1 4 2 3 c. − − − − − 6 6 3 2 5 2 1 2 2 d. − − − 0 0 4 4 4 10 0 3 5 e. − − 2 0 1 2 1 0 4 0 1 f. 1 0 4 2 3 2 4 0 1 3. Verify that A is diagonalisable by computing P -1AP. − = − − = 1 2 2 0 4 0 0 1 3 , P 4 2 5 0 3 0 1 1 0 A 4. Show that each of the following matrices is not diagonalisable. a. ,k 0 0 1 1 k b. − 0 0 2 0 1 4 1 2 1 5. For each of the following matrices find the matrix P that diagonalises A and hence, find P -1AP. a. − − = 20 17 14 12 A b. − = 6 1 1 0 A c. = 0 1 1 0 1 1 1 0 0 A d. − = 0 0 3 0 3 0 2 0 2 A