MAA 111 Algebra for Science Students [Aljabar untuk Pelajar Sains]

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1 UNIVERSITI SAINS MALAYSIA Peperiksaan Kursus Semasa Cuti Panjang Sidang Akademik 9/ Jun MAA Algebra for Science Students [Aljabar untuk Pelajar Sains] Duration : hours [Masa : jam] Please check that this examination paper consists of SEVEN pages of printed material before you begin the examination [Sila pastikan bahaa kertas peperiksaan ini mengandungi TUJUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions: Anser all ten [] questions [Arahan: Jaab semua sepuluh [] 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 [MAA ] Find the orthogonal projection of the vector u=(,,) onto the subspace of spanned by the vectors v =(,,) and v =(,,) by using a least squares solution Can e find the orthogonal projection using Gram-Schmidt technique? If yes, describe the technique [ marks] Find a subset of the vectors that form a basis for the space spanned by the vectors v,,5,, v,,,, v 4, 5,9,4, v4,4,,, v 5 7,8,, 8, then expressed each vector that is not in a basis as a linear combination of the basis vectors; (a) Find the rank and nullity of the matrix A Determine hether vector u=(-,,,) is orthogonal to the subspace spanned by,,,,,,, and 4,,9, 4 (a) Let H be the set of all vectors of the form t t ;t Find a vector v in such that H=span {v} Does this sho that H is a subspace of Justify your anser? If A is a 5 matrix, then the number of leading s in the reduced ro echelon form of A is at most 4 and the number of parameters in the general solution of Ax= is at most Is this statement true? Justify your anser 5 Find a matrix P that diagonalizes A 5 /-

3 [MAA ] Dapatkan unjuran orthogonal bagi vektor u=(,,) dalam subruang direntangi oleh vektor v =(,,) dan v =(,,) menggunakan penyelesaian kuasa dua terkecil Bolehkan unjuran orthogonal bagi vektor u itu diperolehi menggunakan teknik Gram-Schmidt? Jika ya, terangkan teknik penyelesaiannya [ markah] Dapatkan subset vektor yang membentuk asas bagi ruang yang direntangi oleh vektor-vektor v,,5,, v,,,, v 4, 5,9,4, v4,4,,, v5 7,8,, 8, kemudian dapatkan gabungan linear vektor asas bagi setiap vektor yang bukan asas (a) Dapatkan pangkat dan nuliti bagi matriks A Tentukan sama ada vektor u=(-,,,) adalah ortogonal kepada subruang yang direntangi oleh vektor =(,,,), =(,-,,) dan =(4,,9,) 4 (a) Biar H adalah set vektor dalam bentuk t t ;t Dapatkan suatu vektor v dalam dimana H=rentang {v} Adakah ini menunjukkan H suatu subruang dalam? Jelaskan jaapan anda Jika A adalah matriks 5, maka bilangan lajur paksi dalam baris eselon terturun yang paling tinggi boleh dicapai adalah 4 dan bilangan parameter dalam penyelesaian Ax= yang tertinggi boleh dicapai adalah Adakah pernyataan ini benar? Berikan alasan pada jaapan anda 5 Dapatkan matriks P yang memenjurukan matriks A 5 4/-

4 5 6 (a) Given a matrix A solution set of Ax= 4 [MAA ] ; find A - and hat can you conclude on the a b c Given that d e f 6, find g h i a b c g 4d h 4e i 4f d e f 7 Suppose A,B and C are n n square matrices Indicate hether each statement is true or false (i) (A-B) =A -AB+B, (ii) (AB) =A B (iii) If AB=AC, then B=C (iv) If A is invertible, then A T is invertible and (A T ) - =(A - ) T (v) det(a)=det(a) 8 Let a b a a 4 4 a b be the augmented matrix for a linear system Find for hat values of a and b the system has (a) a unique solution, no solution 9 Find the standard matrix for the linear operator T : that (a) first rotates a vector counterclockise about the x-axis through an angle, then rotates a vector counterclockise about the z-axis through an angle,and then folloed by a contraction ith factor k 4 first made a reflection about the xy-plane, then folloed by a dilation ith factor k 5/-

5 5 [MAA ] 5 6 (a) Diberi matriks A ; dapatkan A - dan apa yang boleh dirumuskan tentang set penyelesaian bagi sistem linear Ax= a b c Diberi d e f 6, dapatkan g h i a b c g 4d h 4e i 4f d e f 7 Jika A,B dan C adalah matriks n n Tentukan sama ada pernyataan berikut benar atau palsu (i) (A-B) =A -AB+B, (ii) (AB) =A B (iii) Jika AB=AC, maka B=C (iv) Jika A tersongsangkan, maka A T juga tersongsangkan dan (A T ) - =(A - ) T (v) penentu(a)=penentu(a) 8 Diberi a b a a 4 4 a b sebagai matriks imbuhan untuk suatu system linear Dapatkan nilai a and b untuk menjadikan (a) sistem ini mempunyai penyelesaian unik, sistem ini tiada penyelesaian 9 Cari matriks asas untuk operator linear T : di mana (a) bermula dengan putaran vektor ikut laan arah jam pada satah-x di, kemudian diikuti putaran vektor ikut laan arah jam pada satah-z di sudut k,dan akhirnya diikuti dengan pengecilan imej dengan faktor 4 bermula dengan refleksi pada satah-xy kemudian diikuti dengan pembesaran imej vektor pada faktor k 6/-

6 6 [MAA ] (a) Given a linear transformation T : defined by the equations x x x 4x Is T a one-to-one transformation? Justify your anser [8 marks] 7/-

7 7 [MAA ] (a) Diberi transformasi linear persamaan T : yang didefinisikan dalam bentuk x x x 4x Adakah T suatu transformasi satu-ke-satu? Berikan alasan kepada jaapan anda [8 markah] - ooo O ooo -

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