Student
Number
Semester 2 Assessment, 2019
School of Mathematics and Statistics
MAST10007 Linear Algebra
Writing time: 3 hours
Reading time: 15 minutes
This is NOT an open book exam
This paper consists of 6 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• No written or printed materials may be brought into the examination.
• No calculators of any kind may be brought into the examination.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• There are 13 questions on this exam paper.
• All questions may be attempted.
• Start each question on a new page. Clearly label each page with the number of the
question that you are attempting.
• Marks may be awarded for
– Using appropriate mathematical techniques.
– Accuracy of the solution.
– Full explanations, including justification of rules or theorems used.
– Using correct mathematical notation.
• The total number of marks available is 120.
Instructions to Invigilators
• Students must NOT remove this question paper at the conclusion of the examination.
• Initially students are to receive the exam paper and two 11 page script books.
This paper may be held in the Baillieu Library
MAST10007 Semester 2, 2019
Question 1 (9 marks)
Consider the system of equations
x + 2y + 2z = 2
2x + 5y + 3z = 5
x + 3y + k2z = k + 2
where x, y, z ∈ R and k ∈ R.
(a) Determine the values of k, if any, for which the system has
(i) a unique solution, (ii) no solutions, (iii) infinitely many solutions.
(b) Find all solutions to the system when k = 1.
Question 2 (11 marks)
(a) Consider the matrices
(b) Find the eigenvalues and corresponding eigenvectors for A.
(c) Find an invertible matrix P and a diagonal matrix D such that A = PDP−1.
(d) Assuming that today is fine we have r0 = 0 and f0 = 1. Find formulas for rn and fn for
n ≥ 1.
(e) What are the long term probabilities of rainy days rn and fine days fn, as n→∞?
Question 11 (10 marks)
Consider R3 with the standard inner product given by the dot product
〈u,v〉 = u · v = u1v1 + u2v2 + u3v3.
Let W ⊂ R3 be the subspace spanned by
{(0, 1, 1), (1, 0, 1)}.
(a) Find an orthonormal basis for W .
(b) For v = (1, 1, 0) ∈ R3, find
(i) the orthogonal projection of v onto W ,
(ii) the distance from v to W .
Page 5 of 6 pages
MAST10007 Semester 2, 2019
Question 12 (7 marks)
(a) Find the least squares line of best fit y = a+ bx for the data points
{(−1, 2), (0, 1), (1, 2), (2, 3)}
(b) Draw a clear graph showing the data points and your line of best fit.
Question 13 (4 marks)
Let A be an n× n real matrix. Fix a real number λ and consider the set
w ∈ Rn | (A− λI)2w = 0} .
Show that W 6= {0} if and only if λ is an eigenvalue of A.
End of Exam—Total Available Marks = 120
Page 6 of 6 pages