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代写MAST10007帮写回归、asp实验作业代写

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 
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