QBUS1040: Foundations of Business Analytics Assignment 3 Semester 2, 2023
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QBUS1040: Foundations of Business Analytics
Assignment 3
Semester 2, 2023
Due: 11:59pm on Tuesday, the 17th October (Sydney time)
This homework consists of six questions that require you to submit a written response.
You should submit a PDF to GradeScope and match the page number with the questions you answered. You can find detailed instructions on scanning and submitting your assignments on Canvas. If you fail to match the page to the corresponding question, the marker will not be able to view your response, and thus, you will be awarded a 0 mark for the question.
You may not use notation, concepts or material from other classes (say, a linear algebra class you may have taken) in your solutions. All the problems can be done using only the material from this class, and we will deduct points from solutions that refer to outside material. You must
show
all
of
your
working.
Your solutions should be neat. We will deduct points from poorly written solutions even if they are correct.
Late homework will not be accepted unless a special consideration was obtained.
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Question: |
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Points: |
6 |
7 |
4 |
3 |
4 |
1 |
25 |
(a) (3 points) Consider the function f(x1 , x2 ) = x1 cos(x2 ), where x2 is given in radians.
Find the Taylor approximation f(ˆ) at the point z = (1, π/6). Calculate f(x) and f(ˆ)(x)
at x = (2, π/3). Give your answer to 2 decimal places.
(b) (3 points) Now consider the function f : R2 → R2 , which maps from polar co- ordinates to cartesian co-ordinates:
f(x1 , x2 ) = !x1sin(x1cos)
x2(x2)
" ,
where x2 is given in radians.
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Find the Taylor approximation f(ˆ) at the point z = (1, π/6). Calculate f(x) and f(ˆ)(x)
at x = (2, π/3). Give all values to 2 decimal places.
Question 2 Suppose A and B are m × n matrices and ATB = 0.
(a) (3 points) Does this necessarily mean that either A = 0 or B = 0? Either prove that at least one of A or B has to be 0 or give an example of matrices A and B such that A 
0 and B 
0 but ATB = 0.
(b) (4 points) Suppose that B 
0. Explain why this means that the rows of A must be linearly dependent.
Question 3 (4 points) If a scalar α is positive, then its inverse 1/α is also positive. Is the same true for matrices? Consider a matrix A, which contains only positive elements. Either prove that its inverse A− 1 must have only positive elements or give an example of A and A− 1 where A− 1 contains at least one negative element.
Question 4 (3 points) Suppose A is an invertible n × n matrix with inverse A− 1 . Prove that A2 is invertible and give an expression for its inverse (in terms of A and its inverse).
Question 5 Suppose A is a tall matrix with linearly independent columns so that Ax = b is over-determined. Letˆ(x) denote the least squares approximate solution to the set of equations Ax = b. We want to show that $Aˆ(x) − b$ ≤ $b$.
(a) (2 points) First explain why the inequality is true, i.e. $Aˆ(x) − b$ ≤ $b$.
(b) (2 points) When does the inequality hold with equality? i.e. specify the condition required for $Aˆ(x) − b$ = $b$ and explain why it is unique.
Question 6 (1 point) Draw a fabulous motivational unicorn.
Marks awarded based on participation not merit.
2023-10-17