MAT3701: LINEAR ALGEBRA III May/June 2020

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS




MAT3701 May/June 2020

LINEAR ALGEBRA III
Duration : 3 Hours 100 Marks

EXAMINERS :
FIRST : PROF JD BOTHA EXTERNAL : PROF LM PRETORIUS



This is an open book examination. The examination question paper remains the property of the University of South Africa.


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Answer all the questions and show all calculations. Please read the questions carefully.
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2 MAT3701
May/June 2020


QUESTION 1


Let and consider the following subspaces of M2×2 (C) defined by
W1 = {X ∈ M2×2 (C) : AX = XA} and W2 = {X ∈ M2×2 (C) : AX = X}.

(1.1) Find a basis for W1. (8)

(1.2) Find a basis for W1 ∩ W2. (8)

(1.3) Explain whether M2×2 (C) = W1 ⊕ W2. (1)
[17]


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, QUESTION 2

Let T : C3 → C3 be a linear operator such that T2 = T and dim(N(T)) = 2. Show there exists a basis β for C3


such that [ where b,c ∈ C.
[15]


QUESTION 3

Let
1


andw = 00 .

0
Let T : R4 → R4 be the linear operator defined by T(x) = Ax and let W be the T–cyclic subspace of R4 generated by w.

(3.1) Find the T–cyclic basis for W generated by w. (8)

(3.2) Find the characteristic polynomial of TW. (2)

(3.3) For each eigenvalue of TW, find a corresponding eigenvector expressed as a linear combination of the (8) T-
cyclic basis for W.
[18]


QUESTION 4

Consider the inner product space P2 (R) over R with h·,·i defined by
hg,hi = g (a)h(a) + g(b)h(b) + g (c)h(c)
where a, b and c are distinct real numbers. Let β = {fa, fb, fc} be the set of Lagrange polynomials associated with a, b
and c respectively, and let P : P2 (R) → P2 (R) be the orthogonal projection on W = span .

(4.1) Show that ha1fa + b1fb + c1fc,gi = a1g(a) + b1g(b) + c1g(c) for all a1,b1,c1 ∈ R and g ∈ P2(R). (6)

(4.2) Show that is orthonormal. (7)
(4.3) Find a formula for P(g) expressed as a linear combination of β. (7)
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3 MAT3701
May/June 2020


QUESTION 5

Let T : V → V be a linear operator on a finite-dimensional inner product space V over C.

(5.1) Define what is meant by the adjoint operator T∗ of T. (2)

(5.2) Define what is meant by a normal operator T. (2)

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, (5.3) If T is normal, show that kT(v)k = kT∗(v)k for all v ∈ V . (5)

(5.4) Suppose V = C3 and T : C3 → C3 is defined by T (z1,z2,z3) = (z1 + iz2 − iz3,−iz1, iz1 + iz3). Find (9) a formula for T∗
(z1,z2,z3).
[18]



QUESTION 6

It is given that A ∈ M3×3 (C) is a normal matrix with eigenvalues 1 and −1 and corresponding eigenspaces


E1 = span

and

E−1 = span .

(6.1) Find the spectral decomposition of A. (11)

(6.2) Find A. (1)
[12]

TOTAL MARKS: [100]

c

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