MAT 111 Linear Algebra [Aljabar Linear]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination 0/0 Academic Session June 0 MAT Linear Algebra [Aljabar Linear] Duration : hours [Masa : jam] Please check that this examination paper consists of SIX pages of printed materials before you begin the examination [Sila pastikan bahawa kertas peperiksaan ini mengandungi ENAM muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions: [Arahan: Answer all four [4] questions Jawab semua empat [4] soalan] In the event of any discrepancies the English version shall be used [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan versi Bahasa Inggeris hendaklah diguna pakai] /-

2 - - [MAT ] (a) Using properties of matrix algebra expand and simplify the expression B A C C A B B C and then compute it with the following matrices A B and C 0 A B C (i) A be an n n matrix Prove that if A A T A A A then A is symmetric and If B and C are symmetric matrices determine the condition for the product BC to be symmetric also Given x x x 8x 6x 4 x x x 4x 4 x 6x 0x 4x k 4 (i) Find the value of k which makes the system consistent Use the value of k obtained in (i) to find the general solution to the system of equations and express it in parametric vector form 0 A 0 0 Find the inverse of A 0 0 [00 marks] (a) Menggunakan ciri-ciri aljabar matriks kembangkan dan permudahkan ungkapan B A C C A B B C kemudian hitungkannya dengan matriks A B dan C yang berikut 0 A B C T (i) A sebagai suatu matrix n n Buktikan bahawa jika A A A maka A simetrik dan A A Jika matriks B dan C simetrik tentukan syarat supaya hasildarab BC juga simetrik /-

3 - - [MAT ] Diberi x x x 8x 6x 4 x x x 4x 4 x 6x 0x 4x k 4 (i) Cari nilai k yang menjadikan sistem tersebut konsisten Guna nilai k yang diperoleh dalam (i) untuk mencari penyelesaian umum bagi sistem tersebut dan nyatakan dalam bentuk berparameter 0 A 0 0 Cari songsangan bagi A 0 0 [00 markah] (a) Are the vectors u ( 4 ) u (9) and u ( ) linearly independent in among them 4? If so prove it If not write a linear dependence relation V x y z x y z (i) Find the set S such that L ( S) V From the set S obtained in (i) form an orthonormal basis for V by using the Gram-Schmidt process (iii) Explain why S cannot be a basis for W a bx cx P ( ) ab c Prove that W is a subspace of P ( ) or use a counterexample to show that W is not a subspace of P ( ) Given 0 0 B 0 0 (i) Determine the basis for its column space Deduce the rank of B ( B ) from the answer obtained in (i) [00 marks] (a) Adakah vektor u ( 4 ) u (9) dan u ( ) tak bersandar linear dalam 4? Jika ya buktikannya Jika tidak tuliskan suatu hubungan bersandaran linear di kalangan vektor tersebut 4/-

4 - 4 - [MAT ] V x y z x y z (i) Cari set S sedemikian hingga L ( S) V Dari set S yang diperoleh dalam (i) bentukkan suatu asas berortonormal bagi V menggunakan proses Gram-Schmidt (iii) Terangkan mengapa S tidak boleh menjadi asas W a bx cx P ab c Buktikan bahawa W ialah suatu ( ) subruang P ( ) atau gunakan contoh lawan bagi menunjukkan yang W bukan subruang P ( ) Diberi 0 0 B 0 0 (i) Tentukan asas ruang lajurnya Deduksikan pangkat B ( B ) dari jawapan yang diperoleh dalam (i) [00 markah] (a) Show that T : defined by x y T x y x y is not a linear transformation T : and S : be linear transformations defined by x y z T x y 4 zx y z x y S x yx y (i) Is S one to one? Justify your answer Find ImT Is ImT? Justify your answer (iii) By the dimension theorem what is dim Ker T? Is T one-to-one? Is T onto? Justify your answers (iv) Find the standard matrix C for T S such that x y z C x y z T S Given that U is a subspace of with basis ()(4) (i) Find U the orthogonal complement of U Show that U U Suppose that U and W are subspaces of an inner product space V (i) Show that W W If U W then W U [00 marks] /-

5 - - [MAT ] (a) Tunjukkan bahawa T : yang tertakrif oleh x y T x y x y bukan suatu transformasi linear T : dan S : transformasi linear yang tertakrif oleh x y z T x y 4 zx y z x y S x yx y (i) Adakah S satu-ke-satu? Justifikasikan jawapan anda Cari ImT Adakah ImT? Justifikasikan jawapan anda (iii) Melalui teorem dimensi apakah dim Ker T? Adakah T satu-ke satu? Adakah T menyeluruh? Justifikasikan jawapan anda (iv) Cari matriks piawai C bagi T S sedemikian hingga x y z C x y z T S Diberi U adalah subruang dengan asas ()(4) (i) Cari U pelengkap berortogon bagi U Tunjukkan bahawa U U Andai U dan W adalah subruang dari suatu ruang hasildarab terkedalam V (i) Tunjukkan bahawa W W Jika U W maka W U [00 markah] 4 (a) Find the equation y mx c of the least squares line that best fits the data points (00) (-) and (-4) 4 T : and 4 v v v v 4 is some basis of e e e e e is the standard basis of and 4 Suppose e T 0 e T v e T v v e4 T v 6v 8v and e T 4v 7v 9v (i) Find the matrix T Explain why x x x x4 x x x x x4 x (iii) Verify your answer in (i) by showing that x x x x x T x x x x x T 4 4 (i) Prove that if an n n matrix A is non-singular then 0 is not an eigenvalue of A If A A and is an eigenvalue for A then 6/-

6 - 6 - [MAT ] Given A and the polynomial characteristic equation for A) (i) Find the matrices D C and C - such that n Find A f ( x) x ( x ) is such that f( A ) 0 (the CDC A [00 marks] 4 (a) Cari persamaan y mx c of untuk garislurus kuasadua terkecil yang merupakan penghampiran terbaik bagi titik data (00) (-) dan (-4) 4 T : dan 4 v v v v 4 suatu asas bagi e e e e e asas piawai bagi 4 dan Andai e T 0 e T v e T v v e4 T v 6v 8v dan e T 4v 7v 9v (i) Cari matriks T Terangkan mengapa x x x x4 x x x x x4 x (iii) Tentusahkan jawapan anda dalam (i) dengan menunjukkan bahawa x x x x x T x x x x x T 4 4 (i) Buktikan bahawa jika A ialah suatu matriks n n yang tak singular maka 0 bukan nilai eigen bagi A Jika A A dan ialah nilai eigen bagi A maka Diberi A dan polinomial (persamaan cirian bagi A) f ( x) x ( x ) adalah sedemikian hingga f( A ) 0 (i) Cari matriks D C dan C - sedemikian hingga n Cari A CDC A [00 markah] \ - ooo O ooo -

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