64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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20 84 mRrj iznsk jktfkZ V.Mu eqDr fo'ofo|ky;] bykgkckn vf/kU;kl (Assignment) l= 2012&13 Lukrd dyk dk;ZØe Under Graduate Art Programme fo’k; % xf.kr Subject : Mathematics dkslZ “kh’kZd % Linear Algebra Course Title : Linear Algebra Section - A [k.M & d fo’k; dksM% ;w0th0,e0,e0 Subject Code : UGMM dkslZ dksM % ;w0th0,e0,e0-02 Course Code : UGMM-02 vf/kdre vad % 30 Maximum Marks:30 vf/kdre vad % 18 Max. Marks: 18 uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Long Answer Questions. Answer should be given in 800 to 1000 words. Answer all questions. 1 Define a vector space and given an example of it. Docs there exist a finite vector space over an infinite field 6 osDVj Lisl dks crk;s rFkk ,d mnkgj.k nsaA D;k vuUr QhYM esa ,d QkbukbV osDVj Lisl gksrk gSA 2 Define the rank and nullity of a linear transformation. Verify that the map. 6 2 3 2 T : R ® R given by T ( x1 , x2 ) = ( x1 + x , x1 - x2 , x2 ) is linear. Find the rank and nullity of T. 2 3 js[kh; VªWlQkesZ'ku dk U;wfyVh ,oa jSad crk;sA crk;s fd eSi T : R ® R 2 given by T ( x1 , x2 ) = ( x1 + x , x1 - x2 , x2 ) js[kh; gSA T dk U;wfyVh ,oa jSad crk;saA 3 Reduce the following matrix to the normal form and hence find its rank. fuEu eSfVªDl dks ukeZy QkeZ esa fjM~;wl djrs gq;s bldh jSad crk;sA é1 2 -1 2 1 ù A = ê ú ê 2 4 1 - 2 3 6 ú êë 3 6 2 - 6 5úû Section - B [k.M & [k vf/kdre vad % 12 Max. Marks: 12 uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Short Answer Questions. Answer should be given in 200 to 300 words. Answer all questions. 4. Let V be a vector space of dimension 100. If the dimensions of the sub spaces W 1 and W 2 of V are 60 and 63 respectively find the maximum and minimum dimensions of W1 Ç W2 3 ekuk fd V ,d osDVj Lis'k gS ftldh foek 100 ;fn nks miLis'k W 1 rFkk W 2 dh foek 60 rFkk 63 gks rks W1 Ç W2 dh U;wure rFkk vf/kdre foek Kkr djsaA 5- Let V and U be vector spaces over the same field F. prove that the linear Transformation. T : V ® U is injective if and only if Ker T = { 0} 3 ;fn V rFkk U nks osDVj Lisl ns rks fl) djsa fd js[kh; VªkUlQkeZ T V ® U Ker T = 0 : esa ,dkadh gksxh ;fn vkSj dsoy ;fn { } 6 By Cayley-Hamilton theorem, find the inverse of the matrix dSyh gSfeYVu izes; ls fuEu eSfVªDl dk inverse Kkr djsa
21 é1 2 3ù A = ê ú ê 0 5 0 3 ú êë 2 4 3úû 3 7- If x , y, z are linearly independent vectors of R , Prove that x + y, y + z and z + x are also linearly independent. Do they from a basic 3 of R , over R(field of real numbers). 3 ;fn 3 x , y, z R , dk Lora= js[kh; osDVj gS fl) djsa%& x + y, y + z rFkk z + x Hkh Lora= js[kh; gksxkA D;k ;s basic of dh QhYM½ ijA 3 R gksxk R ¼okLrfod la[;k
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Page 3 and 4: 3 5. Derive the wave equation for a
Page 5 and 6: nks ;qxfer dq.Mfy;ksa }kjk iw.kZ vo
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Page 19: 19 - ;fn y = (sin 1 x) gks rks fn[k
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Page 27 and 28: 27 4. If ;fn y = x y b + x a then s
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74 8- lgh fodYi dks fVd djsaA ¼d½
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81 efJe
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83 efJe
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87 6. efJeÅeeLeea meneÙelee mesJe
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89 117 Gòej ØeosMe jepeef
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153 OTHER INFORMATION : Volume 2 :
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189 ╪ ≥ Γ ┼ ┘ 96 G≥ ej e
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64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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