SIT292 - Linear Algebra for Data Analysis Investigation Trimester 2, 2022

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SIT292 - Linear Algebra for Data Analysis

Investigation

Trimester 2, 2022

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 following table gives indicative descriptors for each mark.

Mark

Description

5

Complete and correct, includes everything relevant, no misunderstandings apparent

4

Almost all correct but something missing or some small errors (not typos)

3

A few errors but base understanding is okay

2

Some understanding, but a few significant errors

1

Partial understanding demonstrated

0

No relevant understanding shown

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

Mark   Description                                                                                                              

5       Exceptional

4       Clear explanations and fluent use of notation

2-3     Overall comprehensible explanations and mostly free of notation errors

1       Lacking in clarity, detail, or significant misunderstandings with regard to notation

0       Very little shown in terms of explanation

 

1. Gram-Schmidt procedure for eigenvectors

Consider the following matrix, A

A =  l 1(2) 「1

1

2

−1

1」

2 

i. Find the eigenvalues and eigenvectors for A.

ii. Starting with the eigenvectors you found in (i), construct a set of orthonormal vectors using

the Gram-Schmidt procedure to obtain an orthogonal matrix.  Ensure to comment or provide justification for any choices you make in following the procedure.

5 marks

2. Orthonormal basis for R3

Use the R code question generator (Question 2) linked in the assignment resources folder with title ‘R Code for Questions 2 and 3’.  You need to use your student ID to generate a set of ordered triples. Change the first line

studentnumber  <-  200000001

so that your own student number is used instead.

i. Does the set of ordered triples generated form a basis for R3? Provide a justification for your response.

ii. Starting with the vectors in (i), use the Gram-Schmidt procedure to construct an orthonormal basis for R3 . Verify that the resulting vectors are orthonormal.

5 marks

3. Subspaces

Use the R code question generator (Question 3) linked in the assignment resources folder to generate a 3 × 4 matrix, which we will denote by A. You will need to use your student ID as you did for Question 2.

The system of equations Ax = 0 has solutions that generate a vector subspace V .

i. Solve the system, and then prove that V is a subspace of R4 .

ii. Give a set of vectors that form a basis for V .

iii. Determine the dimension of V .

5 marks

 

4. Properties of linear spaces

Consider the set of ordered pairs defined on the Cartesian product V = {0, 1, 2} × {0, 1, 2}, i.e. pairs x = (x1 ,x2 ) ∈ V with x1  ∈ {0, 1, 2} and x2  ∈ {0, 1, 2}. We can also refer to the pairs as 2-dimensional vectors.

We define scalar multiplication of elements in V such that for a scalar a ∈ {0, 1, 2}, we have a ⊗ x = (a ⊗ x1 ,a ⊗ x2 ).  Furthermore, vector addition between elements in V is defined such that x ⊕ y = (x1 ⊕ y1 ,x2 ⊕ y2 ). The addition (⊕) and multiplication (⊗) operations for scalars are defined according to the following tables.

 0    1    2

0

1

2

i. Determine whether or not V , along with ⊗ and ⊕ defines a vector space (i.e., verify the 10 properties).

ii. Find a subspace of V with dimension equal to 1, and verify that it is also a vector space.

5 marks

5. Linear codes

The code words

u1 = 110010,u2 = 010100,u3 = 010111,u4 = 101111.

form a basis for a (6,4) linear binary code.

i. Write down a generator matrix for this code.

ii. Construct code words for the messages 1110, 1001 and 0110.

iii. Write down the parity check matrix for this code.

iv. Find syndromes for the received words 101010, 100101 and 001110.

5 marks

2    Section B - Investigation - Choose 1 topic

The aim of this task is to demonstrate that you can independently learn the basics of an advanced topic related to linear algebra. You will be assessed on your ability to convey the central ideas clearly (and in your own words), and on your understanding as demonstrated through original worked example(s) or applications. An example on LU factorisation (which is not one of the available topics) is provided in the assessment resources folder to give you an idea of what a completed task could look like.

There will be 5 criteria, each worth 5 marks, where the mark will be based on the degree to which the criterion is fulfilled.  A score of 3 would indicate that the work is at the standard expected, 4 indicates that the work is above expectations and 5 is exceptional.  Scores of 0 to 2 would indicate the criterion is either not addressed or not addressed to the standard expected.

Criteria

Description

1

Central ideas covered in introduction/summary

2

Concept or process is related back to ideas covered in SIT292

3

Worked Examples and/or example applications demonstrate understanding of the concept or process

4

Quality of Explanation and Presentation

5

References used appropriately and listed according to Deakin’s style guidelines

The task should be the equivalent of approximately 500 words – or 1-3 pages at 11pt fontsize including examples, well set out equations, diagrams etc.   It should be understandable to another student enrolled in this subject and include at least 3 references.

Choose 1 of the following topics for investigation.

Option 1 - Basics of Group Theory

i. Learn the basics of what a‘group’is in group theory and provide a brief introduction in your own words.

ii. Explain how group theory and some related concepts (e.g., Fields, Rings) relate to topics covered in SIT292 and provide original examples demonstrating fulfillment of properties, how the theory is useful etc.

iii. Give an example of how the theory is applied in a different context or with a different mathemat- ical structure, e.g. Dihedral groups in group theory or abstract examples of fields.

Option 2 - Matrix Decompositions

i. Learn how to perform either (choose one only) (a) Singular value decomposition, (b) QR decom- position or (c) QZ decomposition (generalised Schur decomposition - for real matrices), and provide a brief introduction in your own words.

You can find basic descriptions of each type under the ‘Matrix decomposition’ Wikipedia entry, however you will need to find resources that go into more detail and provide better explanation.

ii.  Explain how the decomposition is useful in the context of linear algebra, and relate to specific topics covered in SIT292.

iii. Work through an original example with annotation and explanation to help demonstrate how the method can be used.








2022-10-03