The University of Sydney MATH1002 Linear Algebra
Summer School Assignment 1 2020
This assignment is due by 23:59 on Monday, 27st January 2020. A scanned or typed copy of your so lutions must be uploaded to the Turnitin assignment inbox on the Learning Management System Canvas. If you are submitting after this deadline for any reason, you must use the late submission inbox instead, as the regular inbox will be closed.
Please submit a PDF document scan or convert other formats. It must be legible and correctly oriented, and should include
your SID, but not your name, as assignments will be marked anonymously; your tutorial stream name; and
your tutors name.
Please ensure that your submission was successful, and download a submission receipt.
The School of Mathematics and Statistics encourages some collaboration between students when working on
problems, but students must write up and submit their own version of the solutions.
Your answers should be well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please show all working, and use the notation introduced in lectures. Present your arguments clearly using words of explanation and diagrams where relevant. After all, mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to master.
This assignment is worth 10 of your final assessment for MATH1002.
The marker will give you feedback and allocate an overall letter grade and mark to your assignment using the following criteria:
Mark
Grade
Criterion
5
A
Outstanding and scholarly work, answering all parts of all ques tions correctly, with clear and accurate explanations, and all rel evant diagrams and working. There are at most only minor or trivial errors or omissions.
4
B
Very good work, making excellent progress, but with one or two substantial errors, misunderstandings, or omissions throughout the assignment.
3
C
Good work, making good progress, but making more than two distinct sub stantial errors, misunderstandings, or omissions throughout the assignment.
2
D
A reasonable attempt, but making more than three distinct sub stantial errors, misunderstandings, or omissions throughout the assignment.
1
E
Some attempt, with limited progress made.
0
F
No credit awarded.
1
Assignment Questions:
1. Let Q 3,4,7, R 4,2,5 and S 11,2,0.
a vector orthogonal to QR.
2. Let v 6,1,2, w 5,0,3, and P 9,7,31.
i Find a unit vector u orthogonal to both v and w.
ii Let L be the line in R3 that passes through the point P and is perpendicular to both of the
vectors v and w. Find an equation for the line L in vector form.
iii Find parametric equations for the line L .
3. Solve the following system of linear equations
x1 x2 2×3 x4 3×5 1 2×1 x2 2×3 2×4 6×5 2 3×1 2×2 4×3 3×4 9×5 3
4. Determine for which values of k give i a unique solution ii infinitely many solutions and iii no solutions. Show your working.
i Find QR and RS.
ii Find QR and RS.
iii Find the projection of RS onto QR.
iv Hence, or otherwise, express the vector RS as a linear combination of a vector parallel to QR and
2x 2y 2kz 2 2x 3y 7z 4 kx ky z 1
2 2 4 1 3 4.
1 2 3
ii ShowthatifABAandBABthenAandBareidempotent.
In Questions 5 and 6, you are asked to write Octave code. In any part of the following two questions which asks you to write Octave code, you should provide this code in your assignment. You can provide it as a screenshot, typed or even handwritten as long as the marker can clearly read your code.
6. Let A be the following n n matrix:
n 1 … 1 1 n 1.
111 n
ii Using the technique of row reduction, find A1 for n 3 show your working.
iii Write Octave code to construct the matrix A for any n. You can do this as a function with input n, or otherwise as long as the code allows a user to construct A for any n. Show the Octave output for n 10.
5. A matrix A is idempotent if A2 A. i Show that
is idempotent.
i Find A1 for n 2.
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iv Using Octave, find A1 for n 4, n 5.
v For A1 for each of n 2,3,4,5 show that it can be written as:
c1…1 11c1.
n1 111c
and determine the number c.
vi Prove that the matrix above with the c that you found is the inverse of A for any integer n.
7. In the biographical drama Hidden Figures, human computers perform calculations to help NASA put the first American astronaut into orbit. One of these calculations involves using vector projections to find a set of vectors with a special property. In this question, we will do a higher dimensional analogue of this calculation with the help of Octave.
Consider the following 6 6 matrix conveniently formatted in Octave syntax:
M
i
ii
iii iv
v vi
vii
3 8 45 21 64 52 ; 1 1 4 3 4 11 ; 5 4 14 13 31 62 ; 2 3 22 5 58 52 ; 2 5 23 15 2 21 ; 2 4 23 10 38
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Write Octave code to set the vector vi as the ith row of M. Note that in written text, the vector v1 refers to v1 in Octave code. For example, set the variable v1 to be the vector
3 8 45 21 64 52
Write a simple Octave function which takes as input a vector u and returns the unit vector in the direction of u. Provide this code and use it to compute the unit vector in the direction of v1. Give this vector the name u1.
Write a simple Octave function which takes as input two vectors u and v and returns the projection of v onto u. Provide this code and use it to compute the projection of v2 onto u1.
Use Octave to compute
u2 : v2 Proju1 v2 v2 Proju1 v2
Use Octave to verfiy that u1 and u2 are orthogonal. Give the Octave code and output of your verification.
We define
ui : vi Proju1vi Proju2vi …Projui1vi vi Proju1 vi Proju2 vi . . . Projui1 vi
Use the function you wrote in part ii to help you compute u3 , u4 , u5 and u6 . Provide the Octave code for at least one of these computations.
Use Octave to investigate the angles between each of the vectors u1,u2,u3,u4,u5,u6 Note that due to computer roundoff errors, Octave will not always give exact answers when working over real numbers What is notable about them? Provide the Octave code for at least one of these computations.
3