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Math 210: Linear Algebra Question Paper
Math 210: Linear Algebra
Course:Bachelor Of Education Arts
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION ARTS & SCIENCE / BACHELOR OF SCIENCE GENERAL/BACHELOR OF SCIENCE (COMPUTER SCIENCE) /BACEHOR OF SCIENCE (ECONOMICS AND STATISTICS) /BACHELOR OF ARTS (ECONOMICS AND MATHEMATICS
MATH 210: LINEAR ALGEBRA 1
STREAMS: BED (ARTS) &SCI; BSC (GEN); COMP (SCI); ECON&STAT;
BA (ECON&MATHS) TIME: 2 HOURS
DAY/DATE: TUESDAY 6/8/2013 8.30 AM – 10.30 AM
INSTRUCTIONS:
Answer Question 1 and any Other Two Questions
Adhere to the Instructions on the Answer Booklet
Do Not Write on the Question Paper
Question 1 (compulsory) – (30 Marks)
(a) Consider the following system of linear equations.
x_1+x_2+?4x?_3+?3x?_4=5
?2x?_1+?3x?_2+x_3-?2x?_4=1
x_1+?2x?_2-?5x?_3+?4x?_4=3
Determine whether (i) y=(-10,5,1,2) and (ii) x=(-8,6,1,1) are solutions of the system. [2 marks]
(b) By giving reasons determine whether each of the following equations is linear:
(i) 3x+7y-10yz=16
(ii) x+py+ez=log5
(iii) 3x+ky-8z=32 [3 marks]
(c) Consider the system of linear equations
x+ay=4
ax+9y=b
(i) Determine the values of a for which the system has a unique solution.
[2 marks]
(ii) Find the pairs of values (a,b) for which the system has more than one solution. [3 marks]
(d) (i) Define a homogenous system of equations [1 mark]
(ii) Given a system of linear equations which can compactly be put as A ?x=?b
And whose argumented matrix is [A: ?b], using the concept of the rank [A:b] and rank [A] determine the type of solutions each pair of linear systems of equations will be obtained:
(I) x_1+?2x?_2=1
?-x?_1-?2x?_2=-1 [2 marks]
(II) x_1+?2x?_2=1
?2x?_1+?4x?_2=3 [2 marks]
(e) Evaluate Wroskian w(e^x,e^(x^2 ),e^(x^3 ),0) [3 marks]
(f) (i) Find x,y,z such that A is symmetric, where:
A=[¦(7&-6&2x@y&z&-2@x&-2&5)] [2 marks]
(ii) Using appropriate examples, show that for matrices A and B if AB = 0 doesn’t necessarily imply that A = 0 or B = 0. [2 marks]
(g) (i) Let V be the set of all 2x2 matrices and W be the set of all 2x2 matrices with the determinant equal to zero. Determine if W is a subspace of V or not.
[2 marks]
(ii) Let V=R^3 and s={(1,1,1),(1,2,3),(2,-1,1) }. Determine if (1,-2,5) ?L(s).
[4 marks]
(h) If T:U?V is a linear transformation, show that the range of T is a subspace of V. [2 marks]
Question 2 (20 Marks)
(a) For any u,v,w in R^n and k in R, prove that:
(i) = [1 mark]
(ii) =+ [2 marks]
(iii) =k [2 marks]
(iv) =0 and =0 iff u=0 [1 mark]
(b) If u and v are vectors in V with an inner product, prove that
(i) ?u+v?=?u?+?v? [3 Marks]
(ii) =1/4 ?u+v?^2-1/4 ?u-v?^2 [3 marks]
(c) Determine the rational values of a and b for which the following system has:
(i) No solution
(ii) Unique solution
(iii) Infinitely many solutions.
x_1-?2x?_2+?3x?_3=4
?2x?_1-?3x?_2+?ax?_3=5
?3x?_1-?4x?_2+?5x?_3=b [5 marks]
(d) Find the rank of the matrix
(¦(3&1&2@2&1&1@4&2&2)) [3 marks]
Question 3 (20 Marks)
(a) (i) Give the geometric representation of the following system of equations by solving the equations using Gauss elimination methods.
?-x?_1+?2x?_2-?3x?_3=4
?2x?_1-?4x?_2+?6x?_3=-8
x_1-?2x?_2+?3x?_3=-4 [4 marks]
(ii) Using the concept of elementary product, show that the determinant of a
2x2 matrix A=[¦(a&b@c&d)] is given by det (A)=ad-bc [3 marks]
(b) If A_nxn is a matrix, prove that the following statements are equivalent:
(i) A is invertible
(ii) A ?b, has a unique solution for any ?b
(iii) A ?x= ?0 has a trivial solution only.
(iv) A is row equivalent to In. [7 marks]
(c) By first getting the adjoint, find the inverse of the matrix
A=[¦(2&1&-2@3&2&2@5&4&3)]
Hence solve the following systems of equations:
?2x?_1+x_2-?2x?_3=10
?3x?_1+?2x?_2+?2x?_3=1
?5x?_1+?4x?_2+?3x?_3=4 [6 marks]
Question 4 (20 Marks)
(a) (i) Show that the set of vectors s={2+x+x^2,x-?2x?^2,2+3x-x^2 } is linearly independent in p_2. [3 marks]
(ii) Determine whether the vectors u=(1,1,2),v=(2,3,1)and w=(4,5,5)in R^3 are linearly dependent or not. [4 marks]
(iii) Prove that for the set of vectors S={v_1,v_2,…..v_r } in R^n, if r>n, then S is linearly dependent. [5 marks]
(b) (i) Determine whether the set S={(1,2),(1,-1)} is a basis for R^2. [4 marks]
(ii) Prove any two bases of a vector space have the same number of vectors.
[4 marks]
Question 5 (20 Marks)
(a) Find the basis and dimension of the solution space for the equations:
x_1-?3x?_3+x_3=0
?2x?_1-?6x?_2+?2x?_3=0
?3x?_1-?9x?_2+?3x?_3=0 [6 marks]
(b) Suppose the mapping T:n^2?R^2 is defined by T(x,y)=(x_1+x_2,x_1). Show that T is linear transformation. [4 marks]
(c) Let T:R^3?R^3 be defined by
T(x)=[¦(1&0&5@3&0&15@0&0&0)][¦(x_1@x_2@x_3 )]
Find: (i) Basis for Image of T
(ii) Basis for Kernel of T
(iii) Rank and Nullity of T. [10 marks]
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