Giáo trình đại số tuyến tính ths nguyễn cao đạt, ths nguyễn quốc huy

BQ GIAO DUG VA DAO TAO TRUCJNG DAI HOC DAN LAP CtfU LONG Nguydn Cao Dat - Nguyen Quoc Huy Gido Trinh DAI SO TUYEN TINH TR$NGDAIHOCDA • i-AFCuULONG THUVIEN SodAogky: Vinh Long - 2007 >1 Ldl DE TUA Trifdng Dai hoc Dan lap Cifu Long muon khang dinh la trung tam dao tao nguon nhan lire co trinh cao cho khu vifc dong b^ng song Cii’u Long de phuc vu cho cong cube cong nghiep hoa, hien dai hoa dat nude thi phai nang cao chat lifpng dao tao toan dien Mot nhCrng yeu to yeu la dpi ngu thay co, he thong cac giao trinh va trang thiet bi day hoc Cung vdi cac nganh cac cap toan quoc dang day nhanh tien xay dirng va cung cd vi the xu the hoi nhap, trirdng Dai hoc Dan lap Cdu Long dan tting butfc hoan thien de diTOng dau vdi nhiing thach thud do, gop phan dufa nen giao due dai hoc Viet Nam diing vCTng, ngang tam khu vifc va the gio’i Ke hoach xay du’ng ve mot Bo giao trinh sd dung cho cac giang vien lam tai lieu co ban de giang day, cho sinh vien nghien cdu tham khao hoc tap la nhu cau cap bach cua nha trufing Hu’dng ting ke hoach tren, la Can bo, Giang vien co hdu cua nha trudng, chung tbi bien soan cudn giao trinh sau nhieu.nam giang day va chinh sti'a de hoan thien va gibi thieu vdi cac dong nghiep sinh vien trudng Dai hoc Dan lap Ctiu Long Mong rbng, qua sti dung de giang day va hoc tap, quy thay co dong nghiep va cac anh chi sinh vien dong gop de lan tai ban sau, cudn giao trinh tot hon, dap u’ng dope yeu cau day va hoc cua Tru’bng Hy vong rang, mot khong xa TrUdng Dai hoc Dan lap Chu Long se co day du mot Bo giao trinh hoan chinh dung nha trifdng Tran gidi thieu giao trinh den Quy thay co va cac em sinh vien triTdng Dai hoc Dan lap Ciiu Long Vinh Long, thang 09 nam 2007 Q HIEU TRUdNG ThS Nguyen Cao Dat Chiiomg MA TRAN - DINHTHLfC Ma tran la mot khai niem co ban dai so tuyen tinh Nd dac trOng cho he phoong trinh tuyen tinh, bieu dien vecto bang ma tran toa cung nhif bieu dien phep ddi toa khdng gian vecto Hon ntfa, ma tran cung dung de bieu dien anh xa tuyen tinh, dang song tuyen tinh cung nhif dang toan phoong tren cac khdng gian vecto Trong hng dung, ma tran dung de bieu dien nhieu ddi tOpng khac Do do, choong dau tien nay, ta nghien cdu ma tran nham phuc vu cho cac chaong sau cung nho cho nhieu dng dung sau MA TRAN 1.1 Dinh nghia ma tran Mot bang so hinh chu' nhat gom m dong va n cot all a12 aln a21 a22 a2n \aml am2 amn A= dope goi la mot ma tran cap mxn, ky hieu A = (ajj) A = | aH~| L 'JJmxn mxn hay , a ij chi so hang nam d dong thd i, cot thuf j cua ma tran A Tap hop tat ca cac ma tran cap m x n dope ky hieu la Mmx „ Vdi AeMmxn> mxn mxn, so hang ndm d dong thd i, cot thd j, < i < m , < j < n , cua A dope ky hieu la [A] Vi du Vol A = ky hieu A g M2x3, ™ [Alll: , ta co A la ma tran cap 2x3, i [a]12 2, [Ha [A]21 = 4, □ [A]22 = va [A]23 = Chu y rang viec xb ly bang bang bieu gom nhieu dong, nhieu cot, la mot cong cu quen thuoc dd’i song Chang han, de ghi chu sb liipng ban mot mat hang mot ngay, ta dung mot sb Sb ItYcrng ban n mat hang mot diipc bieu dien bang n sb ma ta goi la mot vecto n - chieu, hay mot ma tran cap x n Sb litqng ban n mat hang m diipc bieu dien bang m vecto n - chieu, hay mot ma tran cap m x n Trong xtf ly anh, mot bdc anh den trang co the bieu dien bbng mot ma tran cac bit 0, Trong thong ke u’ng dung, khao sat mot bien phu thuoc theo k bien doc lap, ngiTbi ta thu thap n bo sb lieu, moi bo sb lieu gom k + sb chi gia tri cua k bien doc lap va gia tri cua bien phu thuoc tiiong u’ng Mot bo sb lieu nhiT vay tao mot ma tran cap n x (k + 1), Giong nhu- cac khai niem khac toan hoc, ma tran co the bleu dien nhieu dbi tapng khac tting bai toan tfng dung cu the Ve mat toan hoc, ta xet mot bleu dien quan ciia ma tran viec khao sat cac he phnong trinh tuyen tinh, mot he thong gom nhieu phirong trinh bac nhat theo nhieu an sb Cu the, xet he phiicmg trinh X -x + -2x + y + z 2y + z 3y + z x, y, z la cac an so can tim (1.1) Vai tro ky hieu ciia cac an x, y, z la khong cd y nghia quyet dinh Chang han, he phtrnng trinh cd the viet lai xi - x2 + x3 -xi + + x3 -2X] + 2x2 3x2 + x3 (1.2) vdi cac an la xT, x2 , x3 Ndi khac di, mot he phtfong trinh tuyen tinh duqc hoan toan xac dinh chi bang cac sb hang di kem theo cac an ma ta goi la cac he so va cac sb hang ve phai ma ta goi la cac he so tg Cu the, mot he phPOng trinh tuyen tinh gom m phi/ong trinh theo n an sb duttc xac dinh bang ma tran cap m x n cac he so' va ma tran cap m x cac he sb tii Chang han, he phuPng trinh (1.1) hay (1.2) dupe xac dinh bdi -1 A = -1 ^-2 z r i va B = II Ngoai ra, ta cd the gom chung hai ma tran lai mot ma tran, goi la ma tran cac he so md rong (hay ma tran bo sung} -1 A = -i ^-2 ' i 2^ hay (A|B) 7) ' -1 -1 -2 1 2" 7/ 1.2 Ma tran bang Hai ma tran A va B dupe goi la bang neu chung cd cung cap va cac sb hang tifcrng tfng cua chung bang tifng ddi mot, nghia la [A] = [B]^ vdi moi i, j Vt du Cho hai ma tran A,B e M2x3 , P Q A= 4^ , B= 2? s 4' 2, Ta co A = B neu va chi neu p = 1, q = va s = □ 1.3 Cac ma tran dac biet i) Ma tran khong : la ma tran ma moi so hang cua nd deu la so Ma tran khong cap m x n dupe ky hieu la 0mxn hay van tdt la Vi du 02x3 - '0