Exam (elaborations)
MAT3701 Exam pack 2024(Linear Algebra III)Questions and answers
- Course
- Institution
MAT3701 Exam pack 2024(Linear Algebra III)Questions and answers With accurate answers and assurance that they are in the exam
[Show more]
Seller
Follow
gabrielmusyoka940
Reviews received
Content preview
MAT3701 EXAMPACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:gabrielmusyoka940@gmail.com
, 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 (gabumusyoka928@gmail.com)
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller gabrielmusyoka940. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.50. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
81239 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now