Assignment - School of Mathematics and Statistics
The University of Sydney
School of Mathematics and Statistics
Assignment
MATH1902: Linear Algebra (Advanced)
Semester 1, 2015
Lecturer: Stephan Tillmann
Instructions to students:
This assignment counts for 5% of your overall assessment for MATH1902.
The assignment has 2 questions, and all questions are weighted equally.
You should submit this assignment by giving it to your tutor in your regular
MATH1902 tutorial in Week 3, Wednesday 18 March.
Your answers should be well written, neat, thoughtful, mathematically concise, and a
pleasure to read. Please show all working, and present your arguments clearly. After
all, mathematics is about communicating your ideas. This is a skill that takes time
and effort to master.
Your tutor will give you feedback and allocate an overall letter grade to your assignment using the following criteria:
A+ : Outstanding and scholarly work, answering all parts of all questions, with
clear and accurate explanations and working, appropriate acknowledgement of
sources (if appropriate), and at most minor or trivial errors or omissions.
A: Very good work, making excellent progress on all questions, but with one or two
substantial errors, misunderstandings or omissions throughout the assignment.
B: Good work, making good progress on all questions, but making more than
two distinct substantial errors, misunderstandings or omissions throughout the
assignment.
C: A reasonable attempt, making substantial progress on most parts of the 2 questions.
D: Some attempt, with substantial progress made on only 1 question.
E: No substantial progress made on any of the 2 questions.
Please turn over for the assignment questions. Enjoy!
c 2015 The University of Sydney
Copyright 1
The University of Sydney
School of Mathematics and Statistics
Assignment
MATH1902: Linear Algebra (Advanced)
Semester 1, 2015
Lecturer: Stephan Tillmann
1. Recall that vectors u, v, w are linearly independent if, for scalars α, β, γ, the equation
αu + βv + γw = 0
only holds when α = β = γ = 0. The vectors are linearly dependent if they are not linearly
independent, that is, there is some choice of scalars α, β, γ, not all zero, for which the
equation holds.
Now suppose that u, v, w are linearly independent and use this hypothesis and the definitions to verify each of the following:
(a) The vectors u − v, v − w, w − u are linearly dependent.
(b) The vectors u − v + w, u + v − w, −u + v + w are linearly independent.
2. On the set R3 = {(a1 , a2 , a3 ) | a1 , a2 , a3 ∈ R} define a new addition by:
(a1 , a2 , a3 ) + (b1 , b2 , b3 ) = ( min(a1 , b1 ), min(a2 , b2 ), min(a3 , b3 ) ),
where min(x, y) = x if x ≤ y and min(x, y) = y if x > y.
For any scalar α ∈ R, we use the usual multiplication
α(a1 , a2 , a3 ) = (αa1 , αa2 , αa3 ).
(a) Compute (7, 11, −4) + (9, −2, 0) and (5, 2, −3) +
1
(−30, 7, 31).
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(b) Determine the set of all (x1 , x2 , x3 ) ∈ R3 satisfying the equation
(5, 2, −3) + (x1 , x2 , x3 ) = (5, 2, −4).
(c) Explore similarities and differences to the usual addition and multiplication by scalars.
In particular, is the new addition commutative or associative? Are the distributive
laws still valid? Either give a proof or a counter-example!
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