MAT 111 Linear Algebra [Aljabar Linear]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination 00/0 Academic Session November 00 MAT Linear Algebra [Aljabar Linear] Duration : hours [Masa : jam] Please check that this examination paper consists of SEVEN pages of printed materials before you begin the examination [Sila pastikan bahawa kertas peperiksaan ini mengandungi TUJUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions: [Arahan: Answer all seven [7] questions Jawab semua tujuh [7] 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 Consider the set A,, 0, 0,, 0,, 0, 0 [MAT ] Is A linearly independent? Justify your answer Can A generate? Give your reason Find a basis for using A Justify your answer [8 marks] Consider the subspace W a, b, c c 0 Find a subset S of such that W L ( S), the linear span of S What is the dimension of W? Give your reason Find the orthogonal complement W of W (d) What is the dimension of W? [6 marks] /-

3 [MAT ] Pertimbangkan set A,, 0, 0,, 0,, 0, 0 Adakah A tak bersandar linear? Jelaskan jawapan anda Bolehkah A menjana? Beri alasan anda Cari asas bagi dengan menggunakan A Jelaskan jawapan anda [8 markah] Pertimbangkan subruang W a, b, c c 0 Cari subset S untuk sedemikian W L ( S), rentangan linear untuk S Apakah dimensi untuk W? Beri alasan anda Cari pelengkap ortogon W untuk W (d) Apakah dimensi untuk W? [6 markah] 4/-

4 4 [MAT ] Consider the following system of linear equations: x y z x y z x y z Write the coefficient matrix A for the above system Find the null space N( A ) of A Then determine the rank ( A ) of A (d) (e) (f) Write the definition of the linear transformation T that has matrix A obtained in as its standard matrix What are the domain and codomain of T? Is T one to one? Give your reason Is T onto? Is the system consistent? Give your reason (g) Does the constant vector,, lie in the image ImT of T? Give your reason [0 marks] 4 Write the augmented matrix for the system x y x y x y By using the Gaussian elimination method, show that the system in has no solution Find the best approximate solution to the system in [ marks] 5/-

5 5 [MAT ] Pertimbangkan system persamaan linear berikut: x y z x y z x y z Tuliskan matriks pekali A bagi system di atas Cari ruang nol N( A ) untuk A Kemudian tentukan pangkat ( A ) untuk A (d) (e) (f) Tuliskan definisi untuk transformasi linear T yang mempunyai matriks A diperoleh di sebagai matriks piawai Apakah domain dan kodomain untuk T? Adakah T satu ke satu? Beri alasan anda Adakah T keseluruh? Adakah sistem ini konsisten? Beri alasan anda (g) Adakah vektor pemalar,, terletak pada imej ImT untuk T? Beri alasan anda [0 markah] 4 Tuliskan matriks imbuhan bagi sistem x y x y x y Dengan menggunakan kaedah penghapusan Gaussian, tunjukkan bahawa system di tidak mempunyai penyelesaian Cari penyelesaian hampiran terbaik untuk sistem di [ markah] 6/-

6 6 [MAT ] 5 Let S,, 0,, 0, be a basis for a subspace W of Use the Gram-Schmidt process to obtain an orthonormal basis B v, v for W Find the scalar product of the vector u 5,, with each of the orthonormal vectors ( v and v ) obtained in part Find u v v u v v [6 marks] 6 Consider the matrix A 4 Find the eigenvalues of A Find all the eigenvectors of A (d) What are the algebraic and geometric multiplicities of each eigenvalue? Is A diagonalizable? Give your reason If yes, find the matrix C such that CAC is diagonal and write the matrix CAC [0 marks] 7 Let 4 5 T : be a linear transformation defined by x, x, x, x, x T x, x, x, x, x Find a basis for the image ImT of T What are the nullity nt ( ) and the rank ( T ) of T? [8 marks] 7/-

7 7 [MAT ] 5 Andaikan S,, 0,, 0, asas bagi suatu subruang W untuk Guna proses Gram-Schmidt untuk memperoleh satu asas ortonormal B v, v bagi W Cari hasil darab scalar antara vektor u 5,, dengan setiap vektor ortonormal ( v and v ) diperoleh dalam bahagian Cari u v v u v v [6 markah] 6 Pertimbangkan matriks A 4 Cari nilai eigen untuk A Cari semua vektor eigen untuk A Apakah kegandaan algebra dan geometri untuk setiap nilai eigen? Adakah A terperpenjurukan? Jika ya, cari matriks C sedemikian CAC adalah terperpenjurukan dan tuliskan matriks CAC [0 marks] 7 Andaikan 4 5 T : satu transformasi linear ditakrif oleh x, x, x, x, x T x, x, x, x, x Cari satu asas bagi imej ImT untuk T Apakah kenolan nt ( ) dan pangkat ( T ) untuk T? [8 markah] - ooo O ooo -

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