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LIA601S- LINEAR ALGEBRA 2 - JAN 2020 |
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eo
NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: BACHELOR OF SCIENCE; BACHELOR OF SCIENCE IN APPLIED MATHEMATICS
AND STATISTICS
QUALIFICATION CODE: 07BSOC; 07BAMS
LEVEL: 6
COURSE CODE: LIA601S
COURSE NAME: LINEAR ALGEBRA 2
SESSION: JANUARY 2020
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/ SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINER:
MR G. TAPEDZESA
MODERATOR:
MR B. OBABUEKI
INSTRUCTIONS
1. Examination conditions apply at all times. NO books, notes, or phones are allowed.
2. Answer ALL the questions and number your answers clearly and correctly.
3. Marks will not be awarded for answers obtained without showing the necessary steps
leading to them (the answers).
4. Write clearly and neatly.
5. All written work must be done in dark blue or black ink.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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QUESTION 1. [34 MARKS]
1.1 Determine whether each of the following mappings T is linear, or not. Justify your answer.
(a) T: R? > R’, where T(z, y) = (3y, 2x, —y).
[5]
(b) T: P, + R?’, where T[p(x)] = [p(0), p(1)].
[5]
(c) T: R® > R’, where T(z, y,z) = (x+1,y+2).
[5]
1.2 Define the following terms as they are used in linear algebra:
(a) The kernel of a linear mapping.
[2]
(b) A singular mapping.
[2]
(c) A one-to-one mapping.
[2]
1.3 Let V be the subspace of C[0,2z] spanned by the vectors 1,sinx,cosz, and let T: V > R?®
be the evaluation transformation on V at the sequence points 0, 7,27. Find
(a) T(1+sinz + cosz).
[2]
(b) ker(T).
[5]
1.4 Let F and G be the linear operators on R? defined by
F(x,y) =(«+y,0) and G(x,y) = (-y,2).
Find formulas defining the following linear operators:
(a) 3F —2G.
(2]
(b) FoG.
[2]
(c) G?.
(2]
QUESTION 2. [28 MARKS]
2.1 Let T’: P, + P» be a linear operator defined by
T (ao + aX + agu*) = ap + a; (3x — 5) + ao(3x — 5)”,
and the basis S = {1,x,2?} for Pp.
(a) Find the matrix representation of T relative to S, and denote it by [T]s.
[7]
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(b) By observing that S' is the standard basis for P2, or otherwise, find the coordinate vector
for p= 1+ 2z + 32? relative to the basis S, and denote it by [p]s.
[2]
(c) Use the transition matrix you obtained in part (a) above and the result in (b) to
compute [T(p)]s.
[4]
(d) Hence, determine T(p) = T(1+ 2x + 32”), again by noting that S is the standard
basis for P).
[2]
2.2 Consider the bases
Sy = {pi,po} = {64+ 32, 104+ 22} and Sp = {qm, qo} = {2, 34+ 22}
for P,, the vector space of polynomials of degree < 1.
(a) Find the transition matrix from S; to Sp and denote it by Ps,-ss,.
(7]
(b) Compute the coordinate vector [p]s,, where p = —4+ 2, and use the transition
matrix you obtained in part (a) above to compute [p]s,.
(6]
QUESTION 3. [20 MARKS]
3.1 Prove that the characteristic polynomial of a 2 x 2 matrix A can be expressed as
d? — tr(A)A + det(A).
[4]
3.2 Suppose A=
0 Q..-2
]1 2 1]
10 8
—2
andP=]1
1
Q -1
1 =O
01
(a) Confirm that P diagonalises A, by finding P~! and computing P~'AP = D.
[9]
(b) Hence, find A’.
(7]
QUESTION 4. [18 MARKS]
4.1 Let x7 Ax be a quadratic form in the variables x1, 29,--- ,t,, and define T: R" > R
by T(x) = x? Ax. Show that T(x + y) = T(x) + 2x? Ay + T(y) and T(cx) = c’T(x),
for any x,y € R" andceER.
[8]
4.2 Find an orthogonal change of variables that eliminates the cross product terms in the
quadratic form
Q(x) = 2} — 2} — 4ay 22 + 42923
and express @ in terms of the new variables.
[10]
END OF QUESTION PAPER