MAT3701 EXAM
PACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:
, lOMoARcPSD|44660598
This paper consists of 3 pages.
ANSWER ALL THE QUESTIONS.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
Open Rubric
, lOMoARcPSD|44660598
2 MAT3701
May/June 2016
QUESTION 1
Let U and W be subspaces of a finite-dimensional vector space V over F such that V = U + W. Prove that
V = U ⊕ W if and only if dim(V ) = dim(U ) + dim(W ).
[8]
QUESTION 2
Let T : M3×3 (C) → M 3×3 (C) be the linear operator over C defined by T (X) = AX, where
0 0 0
A = 1 0 0 ,
0 1 1
and let W be the T –cyclic subspace 3×3
of M(C) generated by A.
(2.1) Find the T –cyclic basis for W. (8)
(2.2) Find the characteristic polynomial W
of. T (4)
(2.3) Explain whether WT is one-to-one. (2)
(2.4) Explain whether WT is onto. (2)
(2.5) For each eigenvalue of W ,Tfind a corresponding eigenvector expressed as a linear combination
(8)of the
T -cyclic basis for W.
[24]
QUESTION 3
0.6 0.3
Let A = .
0.4 0.7
(3.1) Describe the Gerschgorin discs in which the eigenvalues of A lie. (3)
(3.2) Explain whether A is a regular transition matrix. (3)
(3.3) Find lim A m. (8)
m→∞
[14]
QUESTION 4
Given that the system
x − y +z = 2
x+y = 2
x − y +z = 0
3
is inconsistent, find a least squares approximate solution
. in R
[14]
[TURN OVER]
, lOMoARcPSD|44660598
3 MAT3701
May/June 2016
QUESTION 5
Let V = W ⊕ W ⊥, where V is a finite-dimensional
inner product space over F and W is a subspace of
Define
V.
U : V → V by
U(w + w⊥ ) = w − w ⊥
, where w ∈ W and ⊥w∈ W⊥ .
(5.1) Show that U is a linear operator. (5)
(5.2) Show that U is self-adjoint. (6)
(5.3) Show that U is unitary. (6)
(5.4) Show that the eigenvaluesUofare λ = 1 and λ = −1, and find the associated eigenspace
each
of (11)
eigenvalue.
[28]
QUESTION 6
It is given that A ∈ M
3×3 (C) is a normal matrix with eigenvalues 1 and i and corresponding eigenspaces
1 1
E 1 = span (2, 2, 1), (1, −2, 2)
3 3
and
1
E i = span (−2, 1, 2).
3
(6.1) Find the spectral decomposition of A. (11)
(6.2) Find A. (1)
[12]
TOTAL MARKS: [100]
c
UNISA 2016
Downloaded by Gabriel Musyoka ()
PACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:
, lOMoARcPSD|44660598
This paper consists of 3 pages.
ANSWER ALL THE QUESTIONS.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
Open Rubric
, lOMoARcPSD|44660598
2 MAT3701
May/June 2016
QUESTION 1
Let U and W be subspaces of a finite-dimensional vector space V over F such that V = U + W. Prove that
V = U ⊕ W if and only if dim(V ) = dim(U ) + dim(W ).
[8]
QUESTION 2
Let T : M3×3 (C) → M 3×3 (C) be the linear operator over C defined by T (X) = AX, where
0 0 0
A = 1 0 0 ,
0 1 1
and let W be the T –cyclic subspace 3×3
of M(C) generated by A.
(2.1) Find the T –cyclic basis for W. (8)
(2.2) Find the characteristic polynomial W
of. T (4)
(2.3) Explain whether WT is one-to-one. (2)
(2.4) Explain whether WT is onto. (2)
(2.5) For each eigenvalue of W ,Tfind a corresponding eigenvector expressed as a linear combination
(8)of the
T -cyclic basis for W.
[24]
QUESTION 3
0.6 0.3
Let A = .
0.4 0.7
(3.1) Describe the Gerschgorin discs in which the eigenvalues of A lie. (3)
(3.2) Explain whether A is a regular transition matrix. (3)
(3.3) Find lim A m. (8)
m→∞
[14]
QUESTION 4
Given that the system
x − y +z = 2
x+y = 2
x − y +z = 0
3
is inconsistent, find a least squares approximate solution
. in R
[14]
[TURN OVER]
, lOMoARcPSD|44660598
3 MAT3701
May/June 2016
QUESTION 5
Let V = W ⊕ W ⊥, where V is a finite-dimensional
inner product space over F and W is a subspace of
Define
V.
U : V → V by
U(w + w⊥ ) = w − w ⊥
, where w ∈ W and ⊥w∈ W⊥ .
(5.1) Show that U is a linear operator. (5)
(5.2) Show that U is self-adjoint. (6)
(5.3) Show that U is unitary. (6)
(5.4) Show that the eigenvaluesUofare λ = 1 and λ = −1, and find the associated eigenspace
each
of (11)
eigenvalue.
[28]
QUESTION 6
It is given that A ∈ M
3×3 (C) is a normal matrix with eigenvalues 1 and i and corresponding eigenspaces
1 1
E 1 = span (2, 2, 1), (1, −2, 2)
3 3
and
1
E i = span (−2, 1, 2).
3
(6.1) Find the spectral decomposition of A. (11)
(6.2) Find A. (1)
[12]
TOTAL MARKS: [100]
c
UNISA 2016
Downloaded by Gabriel Musyoka ()
