METH APPL MATH I
METH APPL MATH I MATH 601
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sin0 case cos 9 sin 9 CHATER I 10 a LINEAR TRANSFORMATIONS Linear mappings from one vector space to another play an important role in mathe matics This chapter provides an introduction to the theory of such mappings In Section 1 the de nition of a linear transformation is given and a number of examples are presented In Section 2 it is shown that each linear transformation L mapping an ndimensional vector space V into an mdimensional vector space W can be represented by an m x n matrix A Thus we can work with the matrix A in place of the operator L In the case that the linear transformation L maps V into itself the matrix representing L will depend on the ordered basis chosen for V Thus L may be represented by a matrix A with respect to one ordered basis and by another matrix B with respect to another ordered basis In Section 3 we consider the relationship between different matrices that represent the same linear transformation In many applications it is desirable to choose the basis for V so that the matrix representing the linear transformation is either diagonal or in some other simple form DEFINITION AND EXAMPLES In the study of Vector spaces the most important types of mappings are linear trans formations DEFINITION mapping L from a vector space V into a vector space W is said to be a linear transformation or a linear operator if 1 LOW vz aLV1 LV2 for all v1 v e V and for all scalars a and 8 4 Section De nition and Examples I89 If L is a linear transformation mapping a vector space V into W it follows from 1 that 2 Lv1 v2 LV1 LV2 01 I3 I1 and 3 L0lV aLv V Vi 0 Conversely if L satis es 2 and 3 then L0tV1 vz L0tV1 L V2 ozLV1 LV2 Thus L is a linear operator if and only if L satis es 2 and 3 Notation A mapping L from a vector space V into a vector space W will be denoted L V gt W When the arrow notation is used it will be assumed that V and W represent vector spaces Let us now consider some examples of linear transformations We begin with linear operators that map R2 into itself In this case it is easier to see the geometric effect of the operator Linear Transformations on I2 EXAM PLE I Let L be the operator de ned by Lx 3x for each x e R2 Since Lax 3ax oz3x aLx and MK Y 3X y OK 03 LX Ly it follows that L is a linear transformation We can think of L as a stretching by a factor of 3 see Figure 411 In general if a is a positive scalar the linear transformation F x ax can be thought of as a stretching or shrinking by a factor of a 4 EXAM PLE 2 Consider the mapping L de ned by LX 16194 for each x e R2 Thus ifx x1 x2T then Lx x1 0T Ify y1 y2T then 0051 ma 1112 yz ax yl I 90 Chapter 4 Linear Transformations xi axis 10 3x an axis x LOO X181 FIGURE 4ll FIGURE 4I2 and it follows that Lwx y 0m yoei x1e1 y1e1 aLX Ly Thus L is a linear transformation We can think of L as a projection onto the x1 axis see Figure 412 EXAMPLE 3 Let L be the operator de ned by Leo x1 x2T for each x x1 x2T in R2 Since an yi axz yz 15 a V x2 aux Buy Max y 13 Y1 2 V it follows that L is a linear transformation The operator L has the effect of re ecting vectors about the x1 axis see Figure 413 4 FIGURE 4J3 X 11 xzT x axis th x 2 Section De nition and Examples I9 Lx x2 m T XX1Xz FIGURE 4l4 EXAMPLE 4 The operator L de ned by T LX x2 x1 is linear since W y m yz 0m ISM mm 151 YI 01LX Ly The operator L has the effect of rotating each vector in R2 by 90 in the counter clockwise direction see Figure 414 EXAMPLE 5 Consider the mapping M de ned by Mx xf x 12 Since Mozx aele a2x 12 aMx it follows that aM X g M 01X whenever a lt 0 and x 7E 0 Therefore M is not a linear transformation 4 Linear Operators from Rquot to R EXAMPLE 6 The mapping L R2 gt R1 de ned by Lx5c1x2 I92 Chapter 4 Linear Transformations is a linear transformation since Lcxx y om yr m yz x1x2 My Y2 aLX BLy Q EXAMPLE 7 The mapping L from R2 to R3 de ned by Lx x2xix1 m7 is linear since Lozx axzax1ax1 ax2T otLx and 39 X Y x2 y2X1 ynxi y1x2 y2T x2 x1 x1 xzT y2y1y1 y2T Lx Ly Note that if we de ne the matrix A by A 1 1 then 3 2 Lx XI Ax x1 X for each x e R2 39 4 In general if A is any m x n matrix we can de ne a linear operator L A from Rquot to R by L Ax 39 Ax for each x e Rquot The operator L A is linear since LAaX y Max y OlAX My aLA X LMy Thus we can think of each m X n matrix A as a linear operator from Rquot to R quot Section I De nition and Examples I93 In Example 7 we saw that the operator L could have been de ned in terms of a matrix A In the next section we will see that this is true for all linear operators from Rquot to R quot Linear Operators from V to W If L is a linear operator mapping a vector space V into a vector space W then i L0v 0W where 0V and 0W are the zero vectors in V and W respectively 3 ii If V1 v are elements of V and 011 0quot are scalars then LOtivi d2V2 39 39 anVn a1LV1 a2LV2 anLVn39 iii L v Lv for all v E V Statement i follows from the condition Lotv aLv with a 0 Statement ii can easily be proved by mathematical induction We leave this to the reader as an exercise To prove iii note that 0w L0v LV V LV L V Therefore L v is the additive inverse of Lv that is L v Lv EXAMPLE 8 If V is any vector space then the identity operator I is de ned by Iv v for all v e V Clearly I is a linear transformation that maps V into itself 11 I3V2 0W1 vz OIIV1 IV2 4 EXAMPLE 9 Let L be the mapping from Ca b to R1 de ned by b Lf f fxdx If f and g are any vectors in Ca b then b Laf g f f gx dx b b a fxdx f goodx Lf Lg Therefore L is a linear transformation 4 I94 Chapter 4 Linear Transformations EXAMPLE I0 Let D be the operator mapping C1ab into Cab de ned by Df f the derivative of f D is a linear transformation since D0tff3g 0tfl g39 lt Df Dg The Image and Kernel Let L V gt W be a linear transformation We close this section by considering the effect that L has on subspaces of V Of particular importance is the set of vectors in V that get mapped into the zero vector of W V DEFINITION Let L V gt W be alinear transformation The kernel of L denoted kerL is de ned by kerL v e VlLv 0w 4 P DEFINITION Let L V gt W be a linear transformation and let S be a subspace of V The image of S denoted LS is de ned by LS w e Wlw LV for some v e S The image of the entire vector space LV is called the range of L 4 Let L V gt W be a linear transformation It is easily seen that kerL is a subspace of V and if S is any subspace of V then LS is a subspace of W In particular LV is a subspace of W Indeed we have39the following theorem gt THEOREM 4Il If L V gt W is a linear transformation and S is a subspace of V then i kerL is a subspace of V ii LS is a subspace of W gtProof It is obvious that the kerL is nonempty since W the zero vector of V is in kerL To prove i we must show that kerL is closed under scalar multiplication and addition of vectors If v e kerL and a is a scalar then Lav aLv aow 0w Therefore av e kerL If v1 v E kerL then LV1 V2 LV1 LV2 0w 0W 0W Therefore v1 vz e kerL and hence kerL is a subspace of V The proof of ii is similar LS is nonempty since 0w MW e LS If w e LS then w Lv for some v e S For any scalar a 39 aw aLv Lav Section I De nition and Examples I95 Since av e S it follows that aw LS and hence LS is closed under scalar multiplication If W1 W2 6 LS then there exist v1 vz e S such that Lvl W1 and Lvz W2 Thus W1 W2 LV1 LVz LVl V2 and hence LS is closed under addition 39 EXAM P LE I I Let L be the linear transformation from R2 into R2 de ned by 171 Lx A vector x is in kerL if and only if x1 0 Thus kerL is the onedimensional subspace of R2 spanned by eg A vector y is in the range of L if and only if y is a multiple of e1 Thus LRz is the onedimensional subspace of R2 spanned by e1 4 EXAMPLE I2 Let L R3 gt R2 be the linear transformation de ned by Lx x1 x2 x2 x3gtT and let S be the subspace of R3 spanned by el and eg If x e kerL then xrxz0 and vx2x30 Setting the free variable x3 a we get and hence kerL is the one dimensional subspace of R3 consisting of all vectors of the form al 11T If x e S then x must be of the form a0bT and hence Lx abT Clearly LS R2 Since the image of the subspace S is all of R2 it follows that the entire range of L must be R2 ie LR3 R2 4 EXAMPLE I3 Let D P3 gt P3 be the differentiation operator de ned by Dpx P39X The kernel of D consists39of all polynomials of degree 0 Thus kerD P1 The derivative of any polynomial in P3 will be a polynomial of degree 1 or less Conversely any polynomial in P2 will have antiderivatives in P3 so each polynomial in P2 will be the image of polynomials in P3 under the operator D It follows then that DP3 P2 1 Show that each of the following are linear transformations from R2 into R2 Describe geometrically what each linear transformation accomplishes I96 Chapter Linear Trahsformatibns a Lx x1x2T d Lx x b Lx x 39 e LX ma C Mquot xzx1T 2 Let L be the linear transformation mapping R2 into itself de ned by Lx x1 cosu x2 sinot x1 sinot x2 cos aT Express x1 x2 and Lx in terms of polar coordinates Describe geometrically the effect of the linear transformation 3 Let a be a xed vector in R2 A mapping of the form Lx x a is called a translation Show that if a a 0 then L is not a linear transformation Illustrate geometrically the effect of a translation 4 Let L R2 gt R2 be a linear transformation If L12T 23T and L1 1T 52T determine the value of L7 5T 5 Determine whether the following are linear transformations from R3 into R2 8 Lx xzix3T b Lx 0 0T C Lx 1 x1x2T d Lx 063151 x2T 6 Determine whether the following are linear transformations from R2 into R3 8 Lx x1x21T b LXx1x2x1 2x2T 0 Lx x1 0 0T d LX x1 121612 x22T 7 Determine whether the following are linear transformations from Rn into Rnxn39 39 a LA 2A c LA A I b LA AT d LA A AT 8 Determine whether the following are linear transformations from P2 to P3 a Lpx xpx b Lpx x2 px 39 c Lpx Px xPx xzp oc 9 For each f 6 CW 1 de ne Lf F where FxXftdt 05x51 0 Show that L is a linear transformation from C0 l to C 0 1 Then determine Lequot and Lxz 10 Determine whether the following are linear transformations from C01 into R1 a Lf no c Lf mm fan2 b Lf mo 1 2 d Lf fo mxnzdx 11 j l 03 14 15 Section I De nition and Examples I97 If L is a linear transformation from V to W use mathematical induction to prove that 39 Lam a2V2 wavy a1LV1 w2LV2 anLVn Let v1 vn be a basis for a vector space V and let L1 and L2 be two linear transformations mapping V into a vector space W Show that if L1Vi L2Vi for each i 1 n then L1 L2 ie show that L1V L2v for all v e V Let L be a linear transformation from R1 into R1 and let a Ll Show that Lxax forallxeR Let L be a linear operator mapping a vector space V into itself De ne Lquot n 2 1 recursively by L1 L Lk1v LLquotv for all v e V Show that Lquot is a linear operator on V for each n gt 1 Let L1 U gt V and L2 V gt W be linear transformations and let L L20L1 be the mapping de ned by Lu L2ltL1u for each u e U Show that L is a linear transformation mapping U into W Determine the kernel and range of each of the following linear transformations from R3 into R3 p a Lx x3x2x1T b Lx x1x20T c Lxx1x1x1T Let S be the subspace of R3 spanned by e1 and 2 For each of the linear operators L in Exercise 16 determine LS Determine the kernel and range of each of the following linear transformations from P3 into P3 3 LPx xp39x c LPx P0x 111 b LPx Px P39X Let L V gt W be a linear transformation and let T be a subspace of W The inverse image of T denoted L 1T is de ned by L 1T v e VLv e T Show that L1T is a subspace of V A linear transformation L V gt W is said to be anetoone if Lvi Lvz implies that v1 vz ie no two distinct vectors v1 v in V get mapped into the same vector W E W Show that L is oneto one if and only if kerL 0V I98 Chapter 4 Linear Transformations 21 A linear operator L V gt W is said to map V onto W if LV W Show that the operator L R3 gt R3 de ned by LX 3519 xi X2 x1 x2 X3T maps R3 onto R3 22 Which of the operators de ned in Exercise 16 are onetoone Which map R3 onto R3 23 Let A be a 2 X 2 matrix and let L A be the linear operator de ned by L Ax Ax Show that a L A maps R2 onto the column space of A b If A is nonsingular then L A maps R2 onto R2 24 Let D be the differentiationoperator on P3 and let SPE P3P00 Show that a D maps P3 onto P2 but is not one toone b D S gt P3 is one to one but not onto MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS In Section 1 it was shown that each m x n matrix A de nes a linear transformation L A from Rquot to R where L Ax Ax 39 for each x 6 R5 In this section we will see that for every linear transformation L mapping R into R quot there is an m X n matrix A such that Lx Ax We will also see how any linear operator between nite dimensional spaces can be represented by a39matrix 9 THEOREM 42I If L is a linear operator mapping Rquot into R quot there is an m X 11 matrix A such that 39 Lx Ax for each x E Rquot In fact the j th column vector of A is given by ajLe jl2n gtProof For j 1 n de ne r aj d1ja2j amj Lej Section 2 Matrix Representations of Linear Transformations I99 Let A aij 31327m AM If xx1e1x2e2xe is an arbitrary element of R then LX x1Lex x2Lez anen x1a1x2a2 xnan xl x2 a1a2 an xn Ax 4 We have established that each linear transformation from R into R39quot can be represented in terms of an m x n matrix Theorem 421 tells us how to construct the matrix A corresponding to a particular linear operator L To get the rst column of A see what L does to the rst basis element e1 of Rquot Set a1 Lel To get the second column of A determine the effect of L on 22 and set a Lez and so on Since the standard basis elements e1 e2 en were used for Rquot We refer to A as the standard matrix representation of L Later we will see how to represent a linear operator with respect to other bases EXAMPLE I De ne the operator L R3 gt R2 by Lxx1 x2 x2 x3gtT for each x x1 x2 x3T in R3 It is easily veri ed that L is a linear operator We wish to nd a matrix A such that Lx Ax for each x e R3 To do this we must determine Lel Meg and Leg Le1 L100T Lez L010T 1 Les L001T We choose these vectors to be the columns of A 1 10 A 01 1 200 Chapter 4 Linear Transformations To check the result compute Ax EXAMPLE 2 Let L be the linear transformation mapping R2 into R2 that rotates each vector by an angle 6 in the counterclockwise direction We can see from Figure 421a that e1 is mapped into cos 0 sin 0T and the image of e2 is 39 sin 6 0030 The matrix A representing the transformation will have cos 6 sin0T as its rst column and sin 9 cos 9T as its second column cos 9 sine l A sin 6 cos 6 If x is any vector in R2 then to rotate it counterclockwise by an angle 6 we simply multiply by A see Figure 421b Now that we have seen how matrices are used to represent linear operators from Rquot to R quot we may ask if it is possible to nd a similar representation for linear operators from V into W where V and W are vector spaces of dimension n and m respectively To see how this is done let E V1V2 v be an ordered basis for V and F w1 wz Wm be an ordered basis for W Let L be a linear operator mapping V into W If v is any Vector in V then we can express v in terms of the basis E Vx1V1x2V2xnvn We will show that there exists an m x n matrix A representing the operator L in the sense that Ax y ifand only if Lv y1W1y2W2 yme FIGURE 42I 0 1 sin0 0039 Ax cos 0 sin a l 1 0 39 a 0 1 1 0 x1 AX x2 x M 4 0 1 l x2Jt3 x3 Section 2 Matrix Representations of Linear Transformations 20l The matrix A characterizes the effect of the operator L If X is the coordinate vector of v with respect to E then the coordinate vector of Lv with respect to F is given by LvF AX The procedure for determining the matrix representation A is essentially the same as before For j 1 n let aj alja2j amj be the coordinate vector of LVj with respect to W1 wz Wm LVjale1ll2jw2quot39amjwm lsjsn LCIA aij 31 8 If VI1V112V2xnvn then I Lv L 21 391 n ijLWj i1 n m 2w 2 jl i1 m n I Z Zaijxj W i1 1 Fori 1 m let n Yi Zaijxj j1 Thus yy1y2 vaT AX is the coordinate vector of Lv with respect to w1 wz Wm We have estab lished the following theorem V THEOREM 422 Matrix Representation Theorem If E V1 v2 vn and F w1 wz Wm are ordered bases for vector spaces V and W respec tively then corresponding to each linear transformation L V gt W there is an m X n matrix A such that Lvp AM for each v E V A is the matrix representing L relative to the ordered bases E and F In fact ajLv1F j12n 202 Chapter 4 Linear Transformations VEV wLveW A m xvEe Rquot Axwpe R FIGURE 422 Theorem 422 is illustrated in Figure 422 If A is the matrix representing L with respect to the bases E and F and x vE the coordinate vector of v with respect to E y w p the coordinate vector of w with respect to F then L maps v into w if and only if A maps X into y EXAM P LE 3 Let L be a linear transformation mapping R3 into R2 de ned by MK x1b1 x2 xab2 for each x 6 R3 where 1 1 1 and 112 I 1 1 Find the matrix A representing L with respect to the ordered bases e1e2e3 andb1b21 SOLUTION Lei 1 0h2 L62 0b1 lbz Le3 0b1 2 The ith column of A is determined by the coordinates of Le with respect to b1b7 for i 12 3 Thus A 1 0 0 o 1 1 EXAMPLE 4 Let L be a linear transformation mapping R2 into itself de ned by Lotb1 bz at bi 23112 where b1 b2 is the ordered basis de ned in Example 3 Find the matrix A repre senting L with respect to b1 b2 Section 2 Matrix Representations of Linear Transformations 203 SOLUTION Lb1 M 0b2 L39b2 1 2112 l 1 1 0 2 EXAMPLE 5 The linear operator D de ned by Dp p maps P3 into P2 Given the ordered bases x2 x 1 and x 1 for P3 and P2 respectively we wish to determine a matrix representation for D To do this we apply D to each of the basis elements of P3 Thus A LDx22xO1 Dx 0xl1 D10x 0l In P2 the coordinate vectors for Dxz Dx 01 are 20T 01T 00T respectively The matrix A is formed using these vectors as its columns 2 0 0 A 010 If px ax2 bx c then the coordinate vector of p with respect to the ordered basis of P3 is a b cT To nd the coordinate vector of Dp with respect to the ordered basis of P2 we simply multiply 39 I 21l lt1 Thus Dax2 bx c Zax b 4 In order to nd the matrix representation A for a linear transformation L R gt 39R with respect to the ordered bases E I11 un and F 2 b1 bm we39must represent each vector LIlj as a linear combination of b1 bm The following theorem shows that determining this representation of Luj is equivalent to solving the linear system Bx Luj P THEOREM 423 Let E uh 111 and F b bm be ordered bases for Rquot and R respectively If L Rquot gt R39 is a linear transformation and A is the matrix representing L with respect to E and F then ajB Lu for j1 n whereBb1 bm 204 Chapter 4 Linear Transformations P Proof If A is representing L with respect to E and F then for j 1 n Lllj aljb1 azjbz aquotij B aj The matrix B is nonsingular since its column vectors form a basis for R39quot Hence a Bquot1Luj jln 4 One consequence of this theorem is that we can determine the matrix represen tation of the operator by computing the reduced row echelon form of an augmented matrix The following corollary shows how this is done 3 COROLLARY 424 If A is the matrix representing the linear operator L Rquot gt R quot with respect to the bases Eu1 un and Fb1 bm m Lu1 Lu is IIA then the reduced row echelon form of In VProotZ Let B b1 bm The matrix B Lu1 Lu is row equiv alent to 3 103 IL111 Llln IlB 1Lll1 B391Llln Ia1 an 1 IA 6 EXAMPLE 6 Let L R2 gt R3 be the linear transformation de ned by 0 162361 x21x1 X2T Find the matrix representations of L with respect to the ordered bases 111 ug and b1 b2 b3 where u112gtT u231gtT and b1lt100T b2110T b3lt111T SOLUTION We must compute Lu1 Lu2 and then transform the matrix b1 b2 b3 Lu1 Lu2 to reduced row echelon form Lu1 2 3 1T and Lu2 1 4 N 11121 100 1 3 01134601042 00112 001 12 Section 2 Matrix Representations of Linear Transformations 205 The matrix representing L with respect to the given ordered bases is 1 3 A 4 2 1 2 The reader may verify that Lll1 411 4b2 b3 Lu2 3b1 an 213 4 206 Chapter 4 Linear Transformations Section 2 Matrix Representations of Linear Transformations 207 208 Chapter 4 Linear Transformations 1 Refer to Exercise 1 of Section 1 For each linear transformation L nd a matrix A representing L 2 For each of the following linear transformations L mapping R3 into R2 nd a matrix A such that Lx Ax for every x in R3 V a LX1x2X3Tx1xz0T b Lx1x2x3T x1 x2 e um x2 x3T x2 x1 x3 w x2T Section 2 Matrix Representations of Linear Transformations 209 3 For each of the following linear transformations L from R3 into R3 nd a matrix A such that Lx Ax for every x in R3 a Lx1x2 x3T x3x2x1T b Lx1x2x3Tx1x1 x27x1 x2 x3T C Lx1 362 X3T 2x3 X2 3x1 2x1 X3T 4 Let L be the linear transformation mapping R3 into R3 de ned by Lltx 2x x2 x32x2 x1 x3 2x3 x1 x2T Determine the standard matrix representation A of L and use A to nd Lx for each of the following vectors x a x111T b x 211T c x 532T 5 Find the standard matrix representation for each of the following linear trans formations a L is the linear transformation that rotates each x in R2 by 45 in the clockwise direction b L is the linear transformation re ects each vector X in R2 about the x1 axis and then rotates it 90 in the counterclockwise direction c L doubles the length of x and then rotates it 30 in the counterclockwise 39 tion d L re ects each vector x about the line x x1 and then projects it onto the x1 axis 1 1 0 39 b 1 112 0 b3 1 O l l 6 Let and let L be the linear transformation from R2 into R3 de ned by Mquot xlbi X2b2 x1 x2b3 Find the matrix A representing L with respect to the bases e1 e2 and 131 b2 b3 1 1 1 Y1 1 Y2 1 y 0 1 0 0 7 Let and let I be the identity operator on R3 a Find the coordinates of 1020 1032 and Ie3 with respect to Yb Y2 Y3 b Find a matrix A such that Ax is the coordinate vector of x with respect to y1Y2sY3 219 Chapter Linear Transformations 8 Let y1 y2 y3 be de ned as in Exercise 7 and let L be the linear transformation from R3 into R3 de ned by L61Y1 2Y2 3y3 C1 62 63 261 3y2 202 cay3 a Find a matrix representing L with respect to the ordered basis y Y2 Y3 b For each of the following write the vector x as a linear combination of y1 y2 y3 and use the matrix from part a to determine Lx i x 752T 11 x 321T 111 x 12 3T 9 Let I 0 0 1 1 0 R 0 1 1 o 0 1 11 1 1 The column vectors of R represent the homogeneous coordinates of points in the plane 39 e a Draw the gure whose vertices correspond to the column vectors of R What type of gure is it b For each of the following choices of A sketch the graph of the gure represented by AR and describe geometrically the effect of the linear transformation V i 0 0 315 0 i A 039 0 ii A i 0 0 o 1 o 039 1 1 o 2 iiiA 0 3913 0 01 10 For each of the following transformations from R2 into R2 nd the matrix representation of the transformation with respect to the homogeneous coordinate system a The transformation L that rotates each vector by 120 in the counterclock wise direction b The transformation L that translates each point 3 units to the left and 5 units up 0 The transformation L that contracts each vector by a factor of one third 39 d The transformation that re ects a vector about the y axis and then translates it up 2 units 1 39 Section 2 Matrix Representations of Linear Transformations 2 I 11 Let L be the linear operator mapping P2 into R2 de ned by l 1 Lou lfo 1 x x 110 Find a matrix A such that a Lcv3xA 3 12 The linear operator L de ned by Lpx p x 120 maps P3 into P2 Find the matrix representation of L with respect to the ordered bases x2 x 1 and 2 1 x For each of the following vectors px in P3 nd the coordinates of L39px with respect to the ordered basis 2 1 x a x2 2x 3 b x2 1 c 3x d 4x2 2x 13 Let S be the subspace of C 1 b spanned by equot xequot and xzequot Let D be the differentiation operator of 8 Find the matrix representing D with respect to equot xequot xzex 14 Let L be a linear transformation from R into Rquot Suppose that Lx 0 for some x 7 0 Let A be the matrix representing L with respect to the standard basis e1 e2 en Show that A is singular 15 Let L be a linear operator mapping a vector space V into itself Let A be the matrix representing L with respect to the ordered basis V1 v of V ie n LVj Zaijvij l n Show that A quot is the matrix representing L quot with respgf to v v 16 Let E u uz 113 and F b1 132 where 111 10 1T 112 121T u3 111T and bl39 1 1T b2 2 1T For each of the following linear transformations L from R3 into R2 nd the matrix representing L with respect to the ordered bases E and F a Mquot x31x1T b LXx1x2x1 X3T C LX 2X2 x1T 17 Suppose that L1 V gt W and L2 W gt Z are linear transformations and E F and G are ordered bases for V W and Z respectively Show that if A represents L1 relative to E and F and B represents L2 relative to F and 22 Chapter 4 Linear Transformations G then the matrix C BA represents L2 0 L1 V gt Z relative to E and G Hint Show BAv L2 0 L1vG for all v E V 18 Let V W be vector spaces with ordered bases E and F respectively If L V gt W is a linear transformation and A is the matrix representing L relative to E and F show that a v e kerL if and only if VE e N A b w e LV if and only if WF is in the column space of A SIMILARITY If L is a linear transformation mapping an n dimensional vector space V into itself the matrix representation of L will depend on the ordered basis chosen for V By using different bases it is possible to represent L by different it x n matrices In this section we consider different matrix representations of linear operators and charac terize the relationship between matrices representing the same linear operator Let us begin by considering an example in R2 Let L be the linear transforrna tion mapping R2 into itself de ned by L ammm 2 0 Lel 1 l and MW 1 it follows that the matrix representing L with respect to e1 e2 is 421 If we use a different basis for R2 the matrix representation of L will change If for example we use 1 1 u1 and uz 1 1 for a basis then to determine the matrix representation of L with respect to 111 u2 we must determine Lul Luz and express these vectors as linear combinations of 111 and u We can use the matrix A to determine Lul and Luz 31H2 3 l i Since Section 3 Similarity 2l3 To express these vectors in terms of 111 and u2 we use a transition matrix to change from the ordered basis e e2 to u1 u2 Let us rst compute the transition matrix from u1 u2 to e1 e2 This is simply Ummb14 l 1 The transition matrix from e1 eg to uh u2 will then be 1 2 NIH U 1 NI Nb To determine the coordinates of Lul Luz withrespect to u1 uz we multiply W H sHil Llll 2m 0u2 Lllz 1u1 1112 NI up UTlLu1 ulAul NI NIH ND N U 1Lu2 u lnu2 NIH NIH Thus and the manix representing L with respect to u1 uz 2 1 0 1 How are A and B related Note that the columns of B are 3 B U1A 17U1A 2 U1Au1u2 U lAU B U1Au2 Uquot1Au1 and 1 1 Hence Thus if i B is the matrix representing L with respect to u1 u2 n A Is the matrix representing L with respect to e1 Q 111 b 18 the transmon matUX correspondln t0 the Chan e of baSlS OIIl Ill 2 g g 1 2I4 Chapter 4 Linear Transformations then 1 B U lAU The results that we have established for this particular linear operator on R2 are typical of what happens in a much more general setting We will show next that the same sort of relationship as given in 1 will hold for any two matrix representations of a linear operator that maps an n dimensional vector space into itself gt THEOREM 43 Let E V1 vn and F w1 wn be two ordered bases for a vector space V and let L be a linear operator mapping V into itser Let S be the transition matrix representing the change from F to E If A is the matrix representing L with respect to E and B is the matrix representing with respect to F then B S IAS gtProof Let x be any vector in R and let v x1w1x2wz xwn Let 2 ySx tAy zBx It follows from the de nition of S that y ME and hence vy1quot1 quotquotquotiquotynvn Since A represents L with respect to E and B represents L with respect to F we have t LVE and z LVF The transition matrix from E to F is S4 Therefore 3 S4 Z It follows from 2 and 3 that S IASX s Ay 5 1 z Bx see Figure 431 Thus S39IASx Bx for every x e Rquot and hence S lAS B 4 Another way of viewing Theorem 431 is to consider S as the matrix repre senting the identity transformation I with respect to the ordered bases F w w and E V1 vn Thus if FIGURE 43I y A t s s X B Z Section 3 Similarity 2 I 5 Basis E V V I S S I B 39 F V B V as1s L FIGURE 432 S represents I relative to F and E A represents L relative to E S 1 represents 139 relative to E and F then L can be expressed as a composite operator I o L o I and the matrix repre sentation of the composite will be the product of the matrix representations of the components Thus the matrix representation of l39 o L o I relative to F is S IAS If B is the matrix representing L relative to F then B must equal S lAS see Figure 432 39 V DEFINlTlON Let A and B be n x n matrices B is said to be similar to A if there exists a nonsingular matrix S such that B S IAS 4 Note that if B is similar to A then A S l lBS 1 is similar to B Thus we may simply say that A and B are similar matrices It follows from Theorem 431 that if A and B are n X n matrices representing the same operator L then A and B are similar Conversely suppose that A repre sents L with respect to the ordered basis V1 v and B S lAS for some nonsingular matrix S If wl wquot are de ned by W1 S11v1 521V2 Snlvn W2 S12v1 S22v2 sngvn wn Slnvl 52nv2 39 Snnvn then w1 wn is an ordered basis for V and B is the matrix representing L with respect to w1 wn EXAMPLE I Let D be the differentiation operator on P3 Find the matrix B representing D with respect to 1x x2 and the matrix A representing D with respect to 1 2x 4x2 2 SOLUTION D1010x0 x2 Dx11Ox0x2 Dx2012x0x2 2I6 Chapter 4 Linear Transformations The matrix B is then given by 0 0 B0 0 2 000 Applying D to 1 2x and 4x2 2 we obtain Dl0 102x04x2 2 D2x2102x0 4x2239 D4x2 20142x04x2 2 0 140 0 The transition matrix S corresponding to the change of basis from 1 2x 4x2 2 to 1 x x2 and its inverse are given by 2 O 9 Nb 1 1 s o and 31 0 0 o o Nb 1 See Example7 from Chapter 3 Section 5 The reader may verify that A 2 5 135 EXAMPLE 2 Let L be the linear operator mapping R3 into R3 de ned by Lx Ax where 2 2 0 A 2 1 1 2 Thus the matrix A represents L with respect to e1 e2 33 Find the matrix repre senting L with respect to y1 y2 y3 where 21 mm mm Section 3 Similarity 2 L7 SOLUTION Ly1 M1 0 0n Oyz 0Y3 L012 AY2 Y2 0Y11Y2 0Y3 L013 Al s 4Y3 0y 0Y2 4y3 Thus the matrix representing L with respect to y1 y2 y3 is 000 D010 lt 004 We could have found D using the transition matrix Y y1y2y3 and computing D Y IAY This was unnecessary due to the simplicity of the action of L on the basis y1 y2 y3 In Example 2 the linear operator L is represented by a diagonal matrix D with respect to the basis y1y2 y3 It is much simpler to work with D than with A For example it is easier to compute Dx and Dquotx than Ax and Anx Generally it is desirable to nd as simple a representation as possible for a linear operator In particular if the operator can be represented by a diagonal matrix that is usually the preferred representation The problem of nding a diagonal representation for a linear operator will be studied in Chapter 6 1 For each of the following linear transformations L from R2 into R2 determine the matrix A representing L with respect to e1 e2 see Exercise 1 of Section 2 and the matrix B representing L with respect to u1 1 1T 112 1 1T 3 L00 xixzT 0 LG X C ME 39 X2xiT d Lx x e Lx M22 2 Let 111 uz and v1 V2 be ordered bases for R2 where Let L be the linear transformation de ned by 39 Lx x1x2T and let B be the matrix representing L with respect to u1u2 from Exer cise 1a 39 a Find the transition matrix S corresponding to the change of basis from 111112 to V1 V2 III a Chapter 4 39Linear Transformations b Find the matrix A representing L with respect to V1 V2 by computing SBsL c Verify that 39 LV1a11V1a21V2 LV2 a12V1 azzvz Let L be the linear transformation on R3 de ned by LX2x1 x2 x3 2152 161 X3 2x3 361 X2T and let A be the standard matrix representation of L see Exercise 4 of Section 2 If u1110Tu2 10 1T and u3 011T then u1u2u3 is an ordered basis for R3 and U u1n2 H3 is the transition matrix corresponding to a change of basis from u1 112 3 to the standard basis e1 e2 e3 Determine the 1matrix B representing L with respect to the basis u1 u2 u3 by calculating U AU Let L be the linear operator mapping R3 into R3 de ned by Lx Ax where 3 1 2 A 2 0 2 2 l 1 39 1 0 V1 1 V2 V3 2 1 39 1 Find the transition matrix V corresponding to a change of basis from V1 V2 V3 to e1 e2 e3 and use it to determine the matrix B representing L with respect to V1 V2 V3 and let ONH Let L be the operator on P3 de ned by Lpx xp39x p X it Find the matrix A representing L with respect to 1 x x2 b Find the matrix B representing L with respect to 1 x 1 x2 c Find the matrix S such that B S lAS d If px a0 alx 021 x2 calculate Lquotpx Let V be the subspace of Cab spanned by 1equotequot and let D be the differentiation operator on V you 5 OI Section 3 Similarity 2 I 9 2 Find the transition matrix S representing the change of coordinates from the ordered basis 1 equot e to the ordered basis 1 coshx sinhx coshx equot e sinhx equot e quot b Find the matrix A representing D with respect to the ordered basis 1 coshx sinhx c Find the matrix B representing D with respect to 1 e e quot d Verify that B S lAS Prove that if A is similar to B and B is similar to C then A is similar to C Suppose that A SAS where A is a diagonal matrix with diagonal elements A1t2 An a Show that AsiAisil n b Show that if x msl 1252 ansquot then Akx alllfsl 0121552 39 ank sn c Suppose that Mil lt l for i 1 n What happens to Akx as k gt oo Explain Suppose that A ST where S is nonsingular Let B TS Show that B is similar to A Let A and B be u x n matrices Show that if A is similar to B then there exist n X n matrices S and T with S nonsingular such that AST and BTS Show that if A and B are similar matrices then detA detB Let A and B be similar matrices Show that a AT and BT are similar b Aquot and Bk are similar for each positive integer k Show that if A is similar to B and A is nonsingular then B must also be nonsingular and A 1 and 3 1 are similar Let A and B be similar matrices and let A be any scalar Show that a A AI and B A1 are similar b detA AI detB AI The trace of an n x n matrix A denoted trA is the sum of its diagonal entries that is trAtl11t122mann Show that a trAB trBA b If A is similar to B then em trB 220 Chapter 4 Linear Transformations 1 N Use MATLAB to generate a matrix W and a vector x by setting W triuones5 and x l 539 The columns of W can be used to form an ordered basis F W1W2W3W4W5 Let L R5 gt R5 be a linear operator such that LW1 W2 LW2 W3 LW3 w4 and MW 4W1 3W2 2W3 W4 LW5 W1 W2 W3 3W4 ws a Determine the matrix A representing L with respect to F and enter it in MATLAB b Use MATLAB to compute the coordinate vector y W lx of x with respect to F c Use A to compute the coordinate vector 1 of Lx with respect to F d W is the transition matrix from F to the standard basis for R5 Use W to compute the coordinate vector of Lx with respect to the stande basis Set A triuones5 gtIlt trilones5 If L denotes the linear operator de ned by Lx Ax for all x in Rquot then A is the matrix representing L with respect to the standard basis for R5 Construct a 5 x 5 matrix U by setting U hanke1ones5 1 1 5 Use the MATLAB function rank to verify that the column vectors of U are linearly independent Thus E u1u2 u3 u4 us is an ordered basis for R5 The matrix U is the transition matrix from E to the standard basis a Use MATLAB to compute the matrix B representing L with respect to E The matrix B should be computed in terms of A U and U 1 b Generate another matrix by setting V toeplitzl 011 1 Use MATLAB to check that V is nonsingular It follows that the column vectors of V are linearly independent and hence form an ordered basis F for R5 Use MATLAB to compute the matrix C which represents L with respect to F The matrix C should be computed in terms of A V and V l c The matrices B and C from parts a and b should be similar Why Explain Use MATLAB to compute the transition matrix S from F to E Compute the matrix C in terms of B S and 8 Compare your result with the result from part b 39 3 In each of the following answer true if the statement is always true and false Chapter Test 22 Let A toeplitz 7 S companones8 1 and set B Squot1 A S The matrices A and B are similar Use MATLAB to verify that the following properties hold for these two matrices a detB detA b BT STATST1 c 31 slAls d B9 s1A9s e B 31 S 1A 305 f detB 31 detA 31 g trB trA Note that the trace of a matrix A can be computed using the MATLAB command trace These properties will hold in general for any pair of similar matrices See Exercises 11 15 of Section 3 otherwise In the case of a true statement explain or prove your answer In the case of a false statement give an example to show that the statement is not always true 1 2 M o P Squot a I Let L Rquot gt Rquot be a linear transformation If Lx1 LX2 then the vectors x1 and x must be equal If L1 and L2 are both linear transformations mapping a vector space V into itself then L1 L2 is also a linear transformation where L1L2 is the mapping de ned by L1 L2v L1v L2v for all v e V If L V gt V is a linear transformation and x e kerL then Lv x Lv for all v E V If L1 rotates each vector x in R2 by 60 and then re ects the resulting vector about ther axis and L2 is a transformation that does the same two operations but in the reverse order then L1 L2 The set of all vectors x used in the homogeneous coordinate system see the apphgzatron on computer graphics and animation in Section 2 forms a subspace of R Let L R2 gt R2 be a linear transformation and let A be the standard matrix representation of L If L2 is de ned by L2x LLx for all x e R2 then L2 is a linear transformation and its standard matrix representation is A2 Let E x1xz x be an ordered basis for R If L1 R 1 Rquot and L2 Rquot gt Rquot have the same matrix representation with respect to E then L1 L2 222 Chapter 4 Linear Transformations 8 Let L Rquot gt Rquot be a linear transformation If A is the standard matrix repre sentation of L then an n x n matrix B will also be a matrix representation of L if and only if B is similar to A 9 Let A B C be n x n matrices If A is similar to B and B is similar to C then A is similar to C 10 Any two matrices with the same trace are similar This statement is the converse of part b of Exercise 15 in Section 3
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