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日期：2022-08-10 09:13

Important notes:

SIT292 - Linear Algebra for Data Analysis Problem Solving Task 1

Trimester 2, 2022

Due: Monday 8 August 2022, 8:00pm (first day of week 5)

• This assignment contributes 20% toward your final grade and for the average student should represent approximately 20 hours of work beyond participation in classes and workshops.

• Your submission can be handwritten but it must be legible.

• Please follow the order of questions in your submission.

• All steps (working) to arrive at your answers must be clearly shown. No marks will be awarded if understanding is not demonstrated.

• Only (scanned) electronic submission will be accepted via the unit site (DeakinSync).

• Your submission must be in ONE pdf file. Multiple files and/or in different file format, e.g. .jpg, will NOT be accepted. It is your responsibility to ensure your file is not corrupted and can be read by a standard pdf viewer.

• These questions must be attempted individually with the submission representing your own work and understanding. You can discuss question approaches and general methods with tutors and other students, including by way of the unit site’s discussion forums, however you should not share drafts of your solutions.

• Students may receive slightly different versions of some questions, so be sure to be aware of this if posting specific questions to the discussion boards.

• You should not post either questions or solutions of the assignment to external websites. This can constitute a serious case of academic misconduct (collusion, plagiarism, contract cheating etc.) and puts you at risk of heavy penalties.

The marking breakdown is as follows.

Section A

Section B

Section C Total

6 Questions - 5 marks each 30 Quality of explanation and use of notation 5

Choose 2 Questions (out of 4) - 5 marks each 10 Quality of explanation and use of notation 5

Choose 1 Question (out of 3) 10 60

Page 1 of 8

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 2 of 8

1 Section A - Complete all problems

Each question is worth 5 marks and will be graded based on correctness of solution, explanation, and understanding demonstrated.

The additional Quality of explanation and use of notation mark will be assessed against the following standards:

 Mark

5

4 2-3 1 0

Description

Exceptional

Clear explanations and fluent use of notation

Overall comprehensible explanations and mostly free of notation errors

Lacking in clarity, detail, or significant misunderstandings with regard to notation Very little shown in terms of explanation

 1. Sets notation

Give an example of 4 non-empty sets, A,B,C and D such that all the following are true:

A∩B̸=∅, C∩D=∅

B ∈ C, A ∈ D

B ⊆ C, A ̸⊂ D

A ⊆ N, B ⊆ N

i. Write the sets out explicitly in terms of the elements, i.e., do not write “A”, “B” as members of any of the sets.

ii. After writing the sets out, show that each of the conditions are satisfied with brief explana- tions where appropriate.

5 marks

2. The power set

i. LetA={1,2,3}andletρbearelationonP(A)×Zthatmapsasettoitscardinality,e.g.

({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A)

What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written

justification as well as the set B itself.

ii. For two non-empty sets E,F, explain whether or not it is possible that both E ∈ F and

E ⊆ F.

iii. For two non-empty sets E, F , explain whether or not it is possible that both E ∈ P (F ) and E ⊆ P(F).

5 marks

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 3 of 8

3. Equivalence relations

Let ρ ⊆ R × R be a relation such that if cos(a) = cos(b) then (a, b) ∈ ρ. i. Verify that ρ is an equivalence relation.

ii. Give an example of 3 elements in one of the equivalence classes.

iii. Consider a database of actors, e.g. on IMDB. Explain whether or not a binary relation S such that (a1, a2) ∈ S if a1 acted in the same movie as a2 is an equivalence relation (comment on all relevant properties). Use examples where appropriate.

5 marks

4. Graphs

i. Draw a connected bipartite graph with 5 labelled vertices, {a, b, c, d, e} = V and 5 edges. Based on the graph you’ve drawn, give the corresponding partition π = {V1, V2} and the relation ρ ⊂ V1 × V2 corresponding with the edges.

ii. Let A be a set of six elements and σ an equivalence relation on A such that the resulting partition is {{a, b, f }, {c, e}, {d}}. Draw the directed graph corresponding with σ on A.

iii. Draw a directed graph with 5 vertices and 10 edges (without duplicating any edges) repre- senting a relation ρ that is reflexive and antisymmetric, but not symmetric or transitive. Note how these properties can be identified from the graph.

5 marks

5. Matrix multiplication and determinants

i. Use the random matrix generator and calculator in module 3.1 under the Matrix multiplication section (and just before Exercise 1) and change the header information so that n_A <- 3 – it should now read:

m_A <- 3

n_A <- 3

m_B <- n_A

n_B <- 4

# number of rows in A

# number of columns in A

# number of rows in B (=ncol in A)

# number of columns in B

Run the code and then, by clearly showing the steps, verify that the first row in AB is equal to

A1,1 × Brow1 + A1,2 × Brow2 + A1,3 × Brow3

ii. Evaluate the following determinant (using minors and cofactors) by first manipulating the matrix so that all elements in the first column are 0 except for the second element (i.e., using property vii in 3.2.4 of the Study Guide). Do this both at the first step (for the 4 × 4 matrix) and for the subsequent 3 × 3 matrix. You can also use properties ii, iii or iv.

5 marks

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 4 of 8

6. Assume that the following determinant calculation is correct.

Use the properties of determinants (3.2.4 in the Study Guide or p30 - p37 of the lecture notes) to evaluate the following determinants. Identify the relevant property(s) and show the required working but do not calculate the determinants directly (i.e. don’t expand using cofactors and minors).

v. In addition to the properties in the Study guide, for the following you can also use the result that the determinant of an upper triangular matrix can be found by multiplying along the diagonal.

111015 3 0 det0 2 2 00 3 2 0

   0 1 5 40 0 −1 4

2301 0002

5 marks

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 5 of 8

2 SectionB-Choose2outof4

Choose 2 of the following problems (if you complete more than 2, only the first two appearing in your work will be assessed). Each question is worth 5 marks and will be graded based on correctness of solution, explanation, and understanding demonstrated.

The Quality of explanation and use of notation mark will be assessed against the same standards as for Section A:

 Mark

5

4 2-3 1 0

Description

Exceptional

Clear explanations and fluent use of notation

Overall comprehensible explanations and mostly free of notation errors

Lacking in clarity, detail, or significant misunderstandings with regard to notation Very little shown in terms of explanation

 1. Factorgrams

Consider the following factorgram for the number 45. Moving right horizontally corresponds with multiplying by 3 and moving upwards corresponds with multiplying by 5.

5 15 45

139

Let A = {1,3,5,9,15,45} and ρ = {(a,b,c) | LCM(a,b) = c}, i.e. c is the lowest common multiple of a and b.

i. Verify and provide a justification that for all a, b ∈ A, it holds that c ∈ A and hence that ρ is a relation on A3.

ii. Let σ1 = {(a,b) | 5a = b} and σ2 = {(a,b) |3a = b}, with σ1,σ2 ⊂ A2. Write out the ordered pairs for σ1, σ2 and use a mapping diagram (e.g. p23 of the online lecture notes) to calculate σ1 ◦ σ2.

iii. State the rule for ordered pairs in σ1 ◦ σ2 and (without determining the ordered pairs explicitly) provide an argument as to whether or not σ1 ◦ σ2 = σ2 ◦ σ1.

iv. Use the relations σ1,σ2 and σ2 to demonstrate a worked example of the third property of compositions given in 1.2.6 of the Study guide (or p24 of the lecture notes - regarding unions of relations).

5 marks

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 6 of 8

2. Hasse diagrams

(a) Consider coordinates in the XY-plane, (x, y) with x, y ≥ 0 and a binary relation ≺ where

(x1,y1) ≺ (x2,y2) if and only if x1 ≤ x2 and y1 ≤ y2. i. Verify that ≺ is a partial order.

ii. Provide an example set A ⊂ R2 such that the following Hasse diagram corresponds with the partial order ≺ on A (from i.) and label the vertices accordingly.

      iii. What element(s), if any, would you need to add to your poset in order for it to be a lattice? Explain why.

5 marks 3. Consider the set T whose elements are all the factors of 1000 and denote a relation τ ⊂ T × T

such that aτb if a is a factor of b.

i. Verify that (T,τ) is a poset.

ii. Give the least upper bound for 8 and 25, as well as one other upper bound (< 1000). iii. Give the greatest lower bound for 20 and 50, as well as any other lower bound (> 1). iv. Is (T,τ) a lattice? Explain why/why not.

5 marks

iii. Determine whether or not the relation is symmetric, transitive, antisymmetric or reflexive, and state whether it is an equivalence relation, poset, or neither. Make any further relevant observations depending on the outcome.

5 marks

4. LetρbearelationonZ+ suchthataρbifthereexistsann∈Zsuchthata×5n =b. i. Give any 3 ordered pairs of the relation such that a, b > 1.

ii. Determine whether or not any of (5, 20), (25, 1), (7, 7) are elements of ρ.

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 7 of 8

3 SectionC-Choose1outof3

Choose 1 of the following problems (if you complete more than 1, only the first submitted will be assessed). Write up your solution to the problem so that it could be understood by someone else completing SIT292, including all required working, investigations and relevant observations. Make connections with the unit material where appropriate and provide references for any external resources used. High quality partial solutions and explorations can also receive high marks.

You will Mark

9-10 7-8

4-6

2-3 0-1

be given a mark out of 10 according to the following standards: Description

Exceptional

Solution or partial solution is clearly presented with sound understanding and insight demonstrated.

Solution demonstrates engagement with the problem,

good understanding and little in the way of misunderstandings or reasoning errors.

Misunderstandings or errors in reasoning apparent in solution provided. Some understanding demonstrated.

Not attempted or little to no understanding demonstrated.

  1. A group of n = 2k people enter a 2-on-2 basketball competition. Before the competition starts, the players are split up into teams of 2. This amounts to partitioning a set A = {A1, A2, . . . , Ak} such that |A1| = ... = |Ak| = 2. Can you find a rule for the number of potential team combinations by looking at the first few cases of n?

10 marks

2. For any partial order there may be a number of linear orders that are consistent. For example,

for the partial order (P (A), ⊂) with A = {1, 2, 3}, the ordering

∅ ≺ {1} ≺ {2} ≺ {1, 2} ≺ {3} ≺ {1, 3} ≺ {2, 3} ≺ {1, 2, 3}

does not violate the partial order, i.e. if a ≺ b → b ̸⊆ a. Find all the consistent linear orders for (P(A),⊂) when A = {1,2,3,4}.

10 marks

3. In the cartoon Futurama, there is an episode where two characters, Amy and Farnsworth, swap their minds in a mind-swapping machine that doesn’t allow the same two bodies to be placed in the machine more than once. They realise, however, that they can bring in other friends to use the machine to help them get all the minds back in the right bodies.

Let A be a 4 × 4 identity matrix representing the situation when everything is normal. The rows can represent the minds of 4 people while the columns represent the bodies and Aij = 1

SIT292 Linear Algebra for Data Analysis Problem Solving Task 1-G2 2022 Tri 2 Page 8 of 8

indicates that the mind of i is in the body of j. Now, let B be a column-swapping matrix that interchanges the first two columns. We then have

10000100 0100 AB=0 1 0 01 0 0 0=1 0 0 0

 00100010 0010

0001 0001 0001

and hence the resulting matrix now shows that the mind of Person 1 (Amy) is in the body of Person 2 (Farnsworth) and vice-versa, with an additional two (Frye and Bender) who haven’t used the mind-swap machine yet and still have their proper minds in their bodies.

i. Identify each of the remaining column-swap matrices that can be used (assuming we are restricted to these 4 characters) and label them as C, D, E, etc.

ii. Is it possible to re-obtain the identity matrix through sequential multiplication of these matrices (without using any column-swap matrix more than once)? If so, identify the sequence. If it is not possible, provide an argument as to why.

* Note, you can use technology to multiply your matrices and do not need to show this working, however you should clearly communicate your problem solving process.

10 marks