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LIA601S - LINEAR ALGEBRA - 1ST OPP - JUNE 2023 |
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nAmlBIA UnlVERSITY
OF SCIEnCE Ano TECHnDLOGY
FACULTYOF HEALTH,NATURALRESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS,STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION:Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATIONCODE: 07BAMS
LEVEL: 6
COURSECODE: LIA601S
COURSENAME: LINEAR ALGEBRA
SESSION:
DURATION:
JUNE 2023
3 HOURS
PAPER:THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
DR. NA CHERE
MODERATOR:
DR. DSIIIYAMBO
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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QUESTION 1 [6]
1.1. If the nullity of the linear transformation T: Pn Mmn is 3, then determine the rank of T. [3]
1.2. Prove that a square matrix A is invertible if and only if Ois not an eigenvalue of A.
[3]
QUESTION 2 [16]
Determine whether each of the following mappings is linear or not.
= 2.1. T: 'F 'F defined by T(f) (f(x)) 2, where 'Fis the vector space of functions on IRL
[5]
= 2.2. T: Mnn Mnn defined by T(A) AC - CA, where C is a fixed n X n matrix.
[11]
QUESTION 3 [11]
T[~]m t-;, m- T[:l T[J LetT: lll.2 lll.3 defined by
= and 1] =
Find and use itto determine
QUESTION 4 [8]
Let 'F be the vector space of functions with basis S = {sint, cost, e- 2t}, and let D: 'F 'F be the
= differential operator defined by D(f(t)) f' (t). Determine the matrix [DJs representing Din the
basis S.
QUESTION 5 [11]
Find the basis and the dimension of the image of L.
QUESTION 6 [11]
Consider the bases B = {1 + x + x 2, x + x2, x2 } and C = {1, x, x2 } of P2 .
6.1. Find the change of basis matrix Ps--c from C to B.
[8]
6.2. Use the result in part (6.1) to compute [p(x)Js where p(x) = 2 + x - 3x2 .
[3]
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QUESTION 7 (26]
[! 0
Consider A=
5
0
-42.]
5
7.1. Write down the characteristic polynomial P(A.) of A and use this to find the eigenvalues of A. [6]
7.2. Find the eigenspaces corresponding to the eigenvalues of A.
(17]
7.3. Is A diagonalizable ? If so, find an invertible matrix P that diagonalizes A.
[3]
QUESTION 8 (11]
Find an orthogonal change of variables that eliminates the cross-product term in the quadratic form
= q(xi, ·x2 , x3 ) 3xf + 2x~ + 4x 1 x 2 and express q in terms of the new variables.
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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