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LIA601S - LINEAR ALGEBRA - 2ND OPP - JULY 2023 |
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nAm I BIA UntVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,NATURAL RESOURCESAND 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:
JULY 2023
3 HOURS
PAPER:THEORY
MARKS: 100
SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
DR. NA CHERE
MODERATOR:
DR. DSI IIYAMBO
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 [12)
For each of the following questions, state whether it is true or false. Justify or give a counter
example if your answer is false.
1.1. The mapping T: lffi.n lffi.ndefined by T(v) = v + v 0 for all v in lffi.nand v 0 a non-zero fixed
vector in lffi.nis linear.
[3]
1.2. A square matrix A is invertible if and only if Ois an eigenvalue of A.
[3]
1.3. If A is an n x n matrix, then the geometric multiplicity of each eigenvalue is less than or
equal to its algebraic multiplicity.
[2]
1.4. lftwo matrices ofthe same size have the same determinant then they are similar.
[4]
QUESTION 2 [35)
2.1. Show that Tis linear.
[12]
2.2. Determine the bases for the kernel and range of T.
[14]
2.3. Is T singular or nonsingular? Explain.
[3]
2.4. State the nullity and rank of T and verify the dimension theorem.
[6]
QUESTION 3 [8]
= = = Let 'B {vi, v2} and C {ui, u2 } be bases for a vector space V and suppose v1 2u 1 + 3u 2
= and v2 Su 1 - 6u 2 .
3.1. Find the change of basis matrix from 'B to C (Pc,__8 ).
[4]
= 3.2. Use the result in part (3.1) to find [x]c for x 3v1 - 2v 2 .
[4]
QUESTION 4 [11)
Consider the bases B = {1 + x + x2 , x + x2 , x2 } and C = {1, x, x2 } of P2 •
4.1. Find the change of basis matrix P8 ,-c from C to B.
[8]
= 4.2. Use the result in part (4.1) to compute [p(x)]s where p(x) 2 + x - 3x 2 .
[3]
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r
QUESTION 5 [27)
= Let T be a linear operator on Jff3i.defined by T(x, y, z) (2x - y - z, x - z, -x + y + 2z).
5.1. Find the matrix of T with respect to the standard basis vectors of Jff3i.•
[6]
5.2. Find the eigenvalues and the corresponding eigenspaces of the linear operator T.
[21]
QUESTION 6 [7]
. [1 3
Find the quadratic form q(xi, x2, x3) for the symmetric matrix A= ; -1
2
END OF SECOND/SUPPLEMENTARY EXAMINATION QUESTION PAPER
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