MAT 111 Linear Algebra [Aljabar Linear]

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1 UNIVERSITI SINS MLYSI First Semester Eamination 2/22 cademic Session Janary 22 MT Linear lgebra [ljabar Linear] Dration : 3 hors [Masa : 3 jam] Please check that this eamination paper consists of SEVEN pages of printed materials before yo begin the eamination. [Sila pastikan bahawa kertas peperiksaan ini mengandngi TUJUH mka srat yang bercetak sebelm anda memlakan peperiksaan ini.] Instrctions: [rahan: nswer all for [4] qestions. Jawab sema empat [4] soalan.] In the eent of any discrepancies, the English ersion shall be sed. [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, ersi ahasa Inggeris hendaklah digna pakai]. 2/-

2 - 2 - [MT ]. (a) Show that the determinant of the matri 2 9k 3k 96. k 2 k 5 is Find ales of real nmber k for which the set is linearly dependent. k 2 k, 5, 3 (c) Determine the redced row echelon form of the matri 4 5. (e) (f) Find the bases for the row space, the colmn space and the nll space of 3 the matri 4 5. State the relationship between the rank and the nllity of the matri in part. Show that the nll space of the colmn space of. T in part is the orthogonal complement of [ marks] 3/-

3 - 3 - [MT ]. (a) Tnjkkan bahawa penent matriks 2 9k 3k 96. k 2 k 5 ialah Dapatkan nilai-nilai nombor nyata k, yang dengannya set k 2 k, 5, adalah bersandar secara linear. 3 (c) Tentkan bentk eselon baris tertrn matriks 4 5. (e) (f) Dapatkan asas rang baris, asas rang lajr dan asas rang nol bagi 3 matriks 4 5. Nyatakan hbngan di antara pangkat dan kenolan matriks dalam bahagian. T Tnjkkan bahawa rang nol dalam bahagian adalah pelengkap berortogon rang lajr. [ markah] 4/-

4 - 4 - [MT ] 2. Let (a) L : be defined by L Show that L is a linear transformation. Is L one to one and onto? Jstify yor answer. (c) Find a basis for kernel of L. Etend the basis in part (c) to obtain a basis for (e) Find a basis for range of L. (f) ased on yor reslt in parts (c), and (e), erify that dim domain L = dim ker L + dim range L. [ marks] 2. ndaikan L : ditakrifkan oleh L (a) Tnjkkan bahawa L adalah sat transformasi linear. dakah L sat dengan sat dan keselrh? Jstifikasikan jawapan anda. (c) Dapatkan asas bagi inti L. Kembangkan asas dalam bahagian (c) ntk memperoleh asas bagi (e) Dapatkan asas bagi jlat L. (f) erdasarkan keptsan yang telah kam perolehi dalam bahagian (c), dan (e), tentsahkan bahawa matra domain L = matra inti L + matra jlat L. [ markah] 5/-

5 - 5 - [MT ] 3. (a) Let and be ectors in (i) Verify that the fnction (, ) satisfies the properties of an inner prodct on Use the definition in part (i) to find the length of 4 State the Cachy-Schwarz ineqality for the Eclidean space 2 with the inner prodct defined in part (i). Choose an ordered basis S for prodct with respect to S. 2 and find the matri of the inner Let W be the set of all matrices of the form a c. Show that W is a sbspace of M, 33 a ector space of all matrices where the operation is the standard addition of matrices and the operation is the standard mltiplication of a matri by a real nmber. d b e [ marks] 3. (a) ndaikan dan adalah ektor dalam (i) Tentsahkan bahawa fngsi (, ) memenhi sifat-sifat hasil darab terkedalam pada Gna takrif dalam bahagian (i) ntk mencari panjang 4 Nyatakan ketaksamaan Cachy-Schwarz bagi rang Eklidan 2 dengan hasil darab terkedalam yang ditakrifkan dalam bahagian (i). Pilih sat asas bertertib S bagi darab terkedalam terhadap S. 2 dan dapatkan matriks hasil 6/-

6 - 6 - [MT ] ndaikan W adalah set kesema matriks berbentk a c. Tnjkkan bahawa W adalah sat sbrang bagi M, 33 sat rang ektor kesema matriks dengan operasi adalah penambahan piawai matriks dan operasi adalah pendaraban piawai matriks dengan nombor nyata. d b e [ markah] 4. (a) Let be an n n matri and let P P be similar to. Show that if is an eigenector of associated with the eigenale of, then P is an eigenector of associated with the eigenale of. Let. (i) Find the characteristic polynomial, the eigenales, and associated eigenectors of. Is similar to a diagonal matri? Jstify yor answer. If yor answer in part is yes, find a nonsinglar matri P sch that P P is diagonal. Is P in part niqe? Jstify yor answer. () Find the eigenales of. (i) Find the eigenales and associated eigenectors of [ marks] 7/-

7 - 7 - [MT ] 4. (a) ndaikan adalah sat matriks n n dan andaikan P P serpa dengan. Tnjkkan bahawa jika adalah ektor eigen yang dikaitkan dengan, nilai eigen, maka P adalah ektor eigen yang dikaitkan dengan, nilai eigen. ndaikan. (i) () (i) Dapatkan polinomial cirian, nilai-nilai eigen, dan ektor-ektor eigen yang berkaitan bagi matriks. dakah serpa dengan sat matriks pepenjr? Jstifikasikan jawapan anda. Sekiranya jawapan anda dalam bahagian adalah ya, dapatkan sat matriks tak singlar P, spaya dengannya P P adalah matriks pepenjr. dakah P dalam bahagian nik? Jstifikasikan jawapan anda. Dapatkan nilai-nilai eigen. Dapatkan nilai-nilai eigen dan ektor-ektor eigen berkaitan bagi [ markah] - ooo O ooo -

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