Master Linear Algebra Concepts with This Comprehensive Exam Guide
School
University of Science and Technology of Hanoi**We aren't endorsed by this school
Course
BIO 123
Subject
Mathematics
Date
Dec 11, 2024
Pages
3
Uploaded by AgentHornet4752
Final Examination: Linear Algebra 2018-2019 University of Science and Technology of Hanoi Groups: D, E, G Date: Monday, January 21, 2019. Duration: 120 minutes. Total marks: 20 Notes, books, calculator and electronic devices are not permitted. You must show your work to get full marks for these questions. 1. Consider the following system of linear equations r—my+32=5 20+ 4y—42=0 T— 2y+2z2=4 (a) (3 marks) For m = 3, solve the system by using Gauss-Jordan elimination method. (b) (1 mark) Find m such that the system has a unique solution. 2. (a) (2 marks) Find the rank of the following matrix: 3 4 1 2 1 4 7 2 A= 1 10 17 1 4 1 3 3 (b) (2 marks) Calculate the determinant of the following matrix: B = — =R — 8 = 8 == 3. Let 7 : R? — R? be a linear map defined by T(x1,x2,x3) = (21 + 229 + 323,421 + Sxo + 63, T + 82 + 923). (a) (2 marks) Find the matrix of T relative to the standard basis of R3. (b) (1 mark) Find a basis for Ker(T). (¢) (1 mark) Find a basis for range(T).
4. Given the following subset of R?: B ={(1,0,1),(1,1,1),(1,2,0)}. (a) (1 mark) Prove that B is a basis for R?. (b) (2 marks) Use the Gram-Schmidt orthormalization process to transform B into an orthonormal basis B” for R, (¢) (1 mark) Find the coordinate matrix of z = (—1, 1, 2) relative to B”. 5. (4 marks) Let 1 1 0 A=11 2 1 0 -1 1 Find all real eigenvalues and corresponding eigenvectors. Is the matrix diagonalizable? If so find a matrix P and D such that A = P~1DP where D is diagonal. If not carefully explain why it is not possible. Final Examination: Linear Algebra 2018-2019 University of Science and Technology of Hanoi Groups: A, B, C, AE Date: Monday, January 21, 2019. Duration: 120 minutes. Total marks: 20 Notes, books, calculator and electronic devices are not permitted. You must show your work to get full marks for these questions. 1. Consider the following system of linear equations 5r1 + 210 + Sr3+ x4 =9 Tl + 2x9 + r3—3r4= —3 3x1 + 629 + (m + 2)z3 — 924 =6 (a) (3 marks) For m = 4, solve the system by using Gauss-Jordan elimination method. (b) (1 mark) Find m such that the system has no solutions. 2. (a) (2 marks) Find the rank of the following matrix: N W = O O ot W ~N W = O Nl ~N W = O (b) (2 marks) Calculate the determinant of the following matrix: B = W N = N = Q —Q =
3. (4 marks) It is known that for any linear transformation 7" from a vector space V to a vector space W we have dim(Ker(7")) + dim(range(7")) = dim(V'). Use this result to prove that there does not exist a linear transformation 7" : R®> — R? such that Ker(T) = {(ml,xz,xg,x4,x5) (S R5 | €T = 3:82,:U3 =214 = x5}. 4. Given the following subspace of R*: U = {(z1,22,23,24) € RY — 21 — 23 + 23 + 4 = 0}. (a) (1 mark) Find a basis for U. (b) (2 marks) Use the Gram-Schmidt orthonormalization process to transform the basis you found in part (a) into an orthonormal basis for U. (¢) (1 mark) Find the orthogonal projection of v = (6,6,6,6) onto U. 5. (4 marks)Let 1 -1 0 A=1[-1 2 1 0 1 1 Do there exist an orthogonal matrix P and a diagonal matrix D such that A = PTDP? Carefully explain why? If these matrices exist then write down a possible P and the corresponding D.