LINEAR ALGEBRA APPLIED


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1 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder Fourier pproimtion of function. REMARK The cross product is defined onl for vectors in R. The cross product of two vectors in R n, n, is not defined here. THE CROSS PRODUCT OF TWO VECTORS IN Here ou will look t vector product tht ields vector in R orthogonl to two vectors. This vector product is clled the cross product, it is most convenientl defined clculted with vectors written in strd unit vector form v v, v, v v i v j v k. Definition of the Cross Product of Two Vectors Let u u e vectors in R i u j u k v v i v j v k. The cross product of u v is the vector R u v u v u v i u v u v j u v u v k. A convenient w to rememer the formul for the cross product u is to use the following determinnt form. i j k u v u u u v Components of u v v v Components of v Technicll this is not determinnt ecuse it represents vector not rel numer. Nevertheless, it is useful ecuse it cn help ou rememer the cross product formul. Using cofctor epnsion in the first row produces u v u v u v i u v u v j u v u v k u v u v i u v u v j u v u v k which ields the formul in the definition. Be sure to note tht the jcomponent is preceded minus sign. LINEAR ALGEBRA APPLIED In phsics, the cross product cn e used to mesure torque the moment M of force F out point A, s shown in the figure elow. When the point of ppliction of the force is B, the moment of F out A is given M AB \ F where AB \ represents the vector whose initil point is A whose terminl point is B. The mgnitude of the moment M mesures the tendenc of AB \ to rotte counterclockwise out n is directed long the vector M. M A AB B F mrk cinotti/shutterstock.com
2 7 Chpter 5 Inner Product Spces TECHNOLOGY Mn grphing utilities softwre progrms cn find cross product. For instnce, if ou use grphing utilit to verif the result of Emple (), then ou m see something similr to the following. Simultion Eplore this concept further with n electronic simultion, for snt regrding specific progrms involving Emple, plese visit Similr eercises projects re lso ville on the wesite. Finding the Cross Product of Two Vectors Let u i j k v i j k. Find ech cross product.. u v. v u c. v v i j k. u v i. i 5j 7k i j k v u i 5j 7k i j j k k Note tht this result is the negtive of tht in prt (). i j k c. v v i j k i j k The results otined in Emple suggest some interesting lgeric properties of the cross product. For instnce, u v v u v v. These properties, long with severl others, re stted in Theorem 5.7. THEOREM 5.7 Algeric Properties of the Cross Product If u, v, w re vectors in R c is sclr, then the following properties re true u v v u u v w u v u w cu v cu v u cv u u u u 6. u v w u v w PROOF The proof of the first propert is given here. The proofs of the other properties re left to ou. (See Eercises 5 57.) Let u v e u u i u j u k v v i v j v k.
3 THEOREM Applictions of Inner Product Spces 7 Then u v is i j k u v u u u v v v u v u v i u v u v j u v u v k v u is i j k v u v v v u u u v u v u i v u v u j v u v u k u v u v i u v u v j u v u v k v u. Propert of Theorem 5.7 tells ou tht the vectors u v v u hve equl lengths ut opposite directions. The geometric impliction of this will e discussed fter estlishing some geometric properties of the cross product of two vectors. Geometric Properties of the Cross Product If u v re nonzero vectors in R, then the following properties re true.. u v is orthogonl to oth u v.. The ngle etween u v is given u v u v sin.. u v re prllel if onl if u v. 4. The prllelogrm hving u v s djcent sides hs n re of u v. θ v Figure 5.8 v sin θ u PROOF The proof of Propert 4 follows. The proofs of the other properties re left to ou. (See Eercises 58 6.) Let u v represent djcent sides of prllelogrm, s shown in Figure 5.8. B Propert, the re of the prllelogrm is Bse Height Are u v sin u v. Propert sttes tht the vector u v is orthogonl to oth u v. This implies tht u v v u is orthogonl to the plne determined u v. One w to rememer the orienttion of the vectors u, v, u v is to compre them with the unit vectors i, j, k, s shown in Figure 5.9. The three vectors u, v, u v form righthed sstem, wheres the three vectors u, v, v u form lefthed sstem. k = i j u v i plne j RightHed Sstems u v This is the plne determined u v. Figure 5.9
4 74 Chpter 5 Inner Product Spces (, 4, ) u Figure 5. v = j + 6k z Figure v 4 4 (,, ) z u v (,, ) u = i + 4j + k The re of the prllelogrm is u v = 6. Find unit vector orthogonl to oth Finding Vector Orthogonl to Two Given Vectors From Propert of Theorem 5.8, ou know tht the cross product i j k u v 4 i j k is orthogonl to oth u v, s shown in Figure 5.. Then, dividing the length of u v, ou otin the unit vector which is orthogonl to oth u v, ecuse Find the re of the prllelogrm tht hs u i 4j k v i j. u v u v u v 4 i j 4 4 k 4, 4, 4, 4, 4, 4, 4,,. u i 4j k v j 6k s djcent sides, s shown in Figure Finding the Are of Prllelogrm From Propert 4 of Theorem 5.8, ou know tht the re of this prllelogrm is u v. Becuse 6 i j k u v 4 6i 8j 6k the re of the prllelogrm is u v squre units.
5 LEAST SQUARES APPROXIMATIONS (CALCULUS) Mn prolems in the phsicl sciences engineering involve n pproimtion of function f nother function g. If f is in C, the inner product spce of ll continuous functions on,, then g is usull chosen from suspce W of C,. For instnce, to pproimte the function f e, ou could choose one of the following forms of g.. g, Liner. g, Qudrtic. g cos sin, Trigonometric Before discussing ws of finding the function g, ou must define how one function cn est pproimte nother function. One nturl w would require the re ounded the grphs of f g on the intervl,, Are f g d to e minimum with respect to other functions in the suspce W, s shown in Figure Applictions of Inner Product Spces 75 f g Figure 5. Becuse integrs involving solute vlue re often difficult to evlute, however, it is more common to squre the integr to otin f g d. With this criterion, the function g is clled the lest squres pproimtion of f with respect to the inner product spce W. Definition of Lest Squres Approimtion Let f e continuous on,, let W e suspce of C,. A function g in W is clled lest squres pproimtion of f with respect to W when the vlue of I f g d is minimum with respect to ll other functions in W. Note tht if the suspce W in this definition is the entire spce C,, then g f, which implies tht I.
6 76 Chpter 5 Inner Product Spces f() = e 4 g() = Figure 5. Finding Lest Squres Approimtion Find the lest squres pproimtion g of f e,. For this pproimtion ou need to find the constnts tht minimize the vlue of I Evluting this integrl, ou hve I f g d e d. e d e e e d e e e e e. Now, considering I to e function of the vriles, use clculus to determine the vlues of tht minimize I. Specificll, setting the prtil derivtives I e I equl to zero, ou otin the following two liner equtions in. e 6 The solution of this sstem is 4e.87 (Verif this.) So, the est liner pproimtion of f( e on the intervl, is g 4e 8 6e Figure 5. shows the grphs of f g on,. Of course, whether the pproimtion otined in Emple 4 is the est pproimtion depends on the definition of the est pproimtion. For instnce, if the definition of the est pproimtion hd een the Tlor polnomil of degree centered t.5, then the pproimting function g would hve een g f.5 f.5.5 e.5 e e.69. Moreover, the function g otined in Emple 4 is onl the est liner pproimtion of f (ccording to the lest squres criterion). In Emple 5 ou will find the est qudrtic pproimtion.
7 5.5 Applictions of Inner Product Spces 77 Finding Lest Squres Approimtion Find the lest squres pproimtion g of f e,. For this pproimtion ou need to find the vlues of,, tht minimize the vlue of I f g d e d e e e 5. g() f() = e Figure 5.4 Setting the prtil derivtives of I with respect to,, equl to zero produces the following sstem of liner equtions The solution of this sstem is 5 9e e.85 6e 6e 57 e.89. (Verif this.) So, the pproimting function g is g Figure 5.4 shows the grphs of f g. The integrl I (given in the definition of the lest squres pproimtion) cn e epressed in vector form. To do this, use the inner product defined in Emple 5 in Section 5.: f, g f g d. With this inner product ou hve I f g) d f g, f g f g. This mens tht the lest squres pproimting function g is the function tht minimizes f g or, equivlentl, minimizes f g. In other words, the lest squres pproimtion of function f is the function g (in the suspce W) closest to f in terms of the inner product f, g. The net theorem gives ou w of determining the function g. THEOREM 5.9 Lest Squres Approimtion Let f e continuous on,, let W e finitedimensionl suspce of C,. The lest squres pproimting function of f with respect to W is given g f, w w f, w w... f, w n w n where B w, w,..., w n is n orthonorml sis for W.
8 78 Chpter 5 Inner Product Spces PROOF To show tht g is the lest squres pproimting function of f, prove tht the inequlit f g f w is true for n vector w in W. B writing f g s f g f f, w w f, w w... f, w n w n ou cn see tht f g is orthogonl to ech w i, which in turn implies tht it is orthogonl to ech vector in W. In prticulr, f g is orthogonl to g w. This llows ou to ppl the Pthgoren Theorem to the vector sum f w f g g w to conclude tht f w f g g w. So, it follows tht f g f w, which then implies tht f g f w. Now oserve how Theorem 5.9 cn e used to produce the lest squres pproimtion otined in Emple 4. First ppl the GrmSchmidt orthonormliztion process to the strd sis, to otin the orthonorml sis B,. (Verif this.) Then, Theorem 5.9, the lest squres pproimtion of e in the suspce of ll liner functions is g e, e, e d e d e d e d 4e 8 6e which grees with the result otined in Emple 4. Finding Lest Squres Approimtion Find the lest squres pproimtion of f sin,, with respect to the suspce W of polnomil functions of degree or less. To use Theorem 5.9, ppl the GrmSchmidt orthonormliztion process to the strd sis for W,,,, to otin the orthonorml sis B w,, w, w, 5 (Verif this.) The lest squres pproimting function g is 6 6. g f, w w f, w w f, w w f ou hve So, g is f, w f, w f, w 5 sin d sin d sin d. Figure 5.5 g g Figure 5.5 shows the grphs of f g
9 FOURIER APPROXIMATIONS (CALCULUS) You will now look t specil tpe of lest squres pproimtion clled Fourier pproimtion. For this pproimtion, consider functions of the form in the suspce W of C, spnned the sis These n vectors re orthogonl in the inner product spce C, ecuse f, g f g d, f g s demonstrted in Emple in Section 5.. Moreover, normlizing ech function in this sis, ou otin the orthonorml sis With this orthonorml sis, ou cn ppl Theorem 5.9 to write The coefficients for g in the eqution re given the following integrls.. f, w n. f, w f, w n f, w n n f, w n,,,..., n,,..., n 5.5 Applictions of Inner Product Spces 79 g cos... n cos n sin... n sin n S, cos, cos,..., cos n, sin, sin,..., sin n. B w, w,..., w n, w n,..., w n cos,..., g f, w w f, w w... f, w n w n. f f g cos... n cos n sin... n sin n f f f cos n, d cos d cos n d sin d sin n d The function g is clled the nthorder Fourier pproimtion of f on the intervl,. Like Fourier coefficients, this function is nmed fter the French mthemticin JenBptiste Joseph Fourier (768 8). This rings ou to Theorem 5.. sin,..., f d f cos d f cos n d sin n. f sin d f sin n d
10 8 Chpter 5 Inner Product Spces THEOREM 5. Fourier Approimtion On the intervl,, the lest squres pproimtion of continuous function f with respect to the vector spce spnned is, cos,..., cos n, sin,..., sin n g cos... n cos n sin... n sin n where the Fourier coefficients,,..., n,,..., n re j j f d f cos j d, j,,..., n f sin j d, j,,..., n. Finding Fourier Approimtion Find the thirdorder Fourier pproimtion of f,. Using Theorem 5., ou hve where g cos cos cos sin sin sin j j d cos j d cos j j j sin j sin j d sin j j j cos j j. This implies tht,,,,,,. So, ou hve f() = g sin sin sin sin sin sin. Figure 5.6 compres the grphs of f g. g ThirdOrder Fourier Approimtion Figure 5.6
11 5.5 Applictions of Inner Product Spces 8 In Emple 7, the pttern for the Fourier coefficients ppers to e..., n,,,..., n n. The nthorder Fourier pproimtion of f is g sin sin sin... n sin n. As n increses, the Fourier pproimtion improves. For instnce, Figure 5.7 shows the fourth fifthorder Fourier pproimtions of f,. f() = f() = g g FourthOrder Fourier Approimtion FifthOrder Fourier Approimtion Figure 5.7 In dvnced courses it is shown tht s n, the pproimtion error f g pproches zero for ll in the intervl,. The infinite series for g is clled Fourier series. f() = g() = + 4 cos + 4 cos 9 Figure 5.8 Find the fourthorder Fourier pproimtion of Finding Fourier Approimtion Using Theorem 5., find the Fourier coefficients s follows. j j cos j j So,, 4,, 49, 4,,,, 4, which mens tht the fourthorder Fourier pproimtion of f is cos j d d g 4 cos 4 cos. cos j d sin j d 9 Figure 5.8 compres the grphs of f g. f,.
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