Sma2110:Real Analysis Ii Question Paper
Sma2110:Real Analysis Ii
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2011
QUESTION ONE (30 MARKS)
a) Compute the minors and cofactors of each element in row 2 of the matrix
=
1 3 0 -2 2 -1 4 0 -2 (4 Marks) b) Let = 4,3 2 and ??1 = 1,3 ,??2 = 2,5 be a basis of 2. Find the coordinate vector of v relative to the basis ?? . (4 Marks) c) Let = 1 2 3 4 and T be the linear mapping from 2 to 2 defined by T(v)=Av, where v is written as a column vector. Find the matrix of T in the basis ??1 = 1,3 ,??2 = 2,5 . (4 Marks) d) Prove that similar matrices have the same determinant. (4 Marks)
e) Compute det A, where ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
??
??
?
21900 32505 84637 00200 53704 A (5 Marks)
f) Show that = 1 4 2 3 is a zero of ?? = 2 - 4 - 5. (5 Marks)
g) Find the characteristic polynomial ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
?
70000 05300 02400 00020 00052 A (4 Marks)
2
QUESTION TWO (20 MARKS)
a) Show that the mapping T:2 3 be defined by ??,?? = ?? + 1,2??,?? + ?? is not a linear mapping. (4 Marks) b) Using the following basis of 3: ??1 = 1,0,0 ,??2 = 0,1,0 ,??3 = (0,0,1) and
??1 = 1,1,1 ,??2 = 1,1,0 ,??3 = (1,0,0) Find the transition matrix P from ?? ?? and Q from ?? to ??. Verify that = -1. (5 Marks) c) Let T:3 2be a linear mapping defined by : ??,??, = (2?? + ?? - ,,3?? - 2?? + 4). Find the matrix T relative to the basis 1,1,1 , 1,1,0 , 1,0,1 of 3 and 1,2 , 2,3 of 2. (5 Marks) d) Let T be the linear mapping defined by : ??,?? = 5?? + ??,3?? - 2?? . Find the matrix of T with respect to the basis ??1 = 1,2 ,??2 = 2,3 . Verify that []??[]?? = () ?? for any vector 2. (6 Marks)
QUESTION THREE (20 MARKS)
a) Let T:2 2 be defined by ??,?? = ?? + ??,-2?? + 4?? . Find the determinant of T. (5 Marks) b) Let T:4 3 be defined by 1,1,1,1 = 1,0,-1 , 0,1,1,1 = 2,1,1 , = 0,0,1,1 = 1,-2,3 ?????? 0,0,0,1 = (-2,3,2). Find (??,??,,) and hence determine (2,3,4,5). (5 Marks) c) Find F(A) when = 1 -2 4 5 and ?? = 2 - 3 + 7. (5 Marks) d) Find the values of t for which the determinant of A is zero given that
=
- 2 4 3 1 + 1 -2 0 0 - 4
(5 Marks)
QUESTION FOUR (20 MARKS)
a) Let F:3 2 be a linear mapping with = 2 5 -3 1 -4 7 as a matrix of f with respect to the standard basis of 3 and 2. Find the matrix of f with respect to the basis = ??1 = 1,1,1 ,??2 = 1,1,0 ,??3 = (1,0,0) of 3 and = 1 = 1,3 ,2 = 2,5 of 2. (6 Marks) b) Prove that similar matrices have the same characteristics polynomial. (3 Marks) c) Let F:3 3 be defined by ??,??, = ?? - ??,2?? + 3?? + 2,?? + ?? + 2 . Find the eigen values and a basis for the eigen space of each eigen value. (5 Marks) d) Find the characteristic polynomial, minimal polynomial and eigen values of the
matrix =
3 -5 5 5 -7 5 5 -5 3 (6 Marks)
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