Faculty of Computer Studies

MT132 – Linear Algebra

Take Home Exam for Final Assignment (Summer 2020/2021)

Cut-Off Date: —

Duration: 48 Hours

Total Marks: 100

Contents

Warnings and Declaration……………………………………………………………… 1

Question 1………………………….…………………………………………………………… 2

Question 2…..………………………………………………………………………………….. 3

Question 3 ………….………..………………………………………………………………… 4

Question 4 ………….………..…………………………………………………………………. 5

Question 5 ………….………..…………………………………………………………………. 6

Plagiarism Warning:

As per AOU rules and regulations, all students are required to submit their own FTHE work and avoid plagiarism. The AOU has implemented sophisticated techniques for plagiarism detection. You must provide all references in case you use and quote another person’s work in your FTHE. You will be penalized for any act of plagiarism as per the AOU’s rules and regulations.

Declaration of No Plagiarism by Student (to be signed and submitted by student with FTHE work):

I hereby declare that this submitted FTHE work is a result of my own efforts and I have not plagiarized any other person’s work. I have provided all references of information that I have used and quoted in my FTHE work.

Name of Student: ………………………………..

Signature: ……………………………………………

Date: ……………………………………………………

Answer the following questions:

Q‒1:

[6+6 marks] Let and.

Find, if exists, .

Find a matrix such that .

[8 marks] Let and be matrices. Assuming that the stated inverses exist, show that .

Q‒2:

[4+4 marks] Let be a subset of , where , and .

Is linearly dependent? Justify your answer.

Is in ? Justify your answer.

[8 marks] Is a linearly independent set of vectors in . Justify your answer.

[4 marks] Determine whether is a subspace of .

Q‒3:

[4+4 marks] Let .

Show that is a subspace of .

Find a basis for .

[12 marks] Find a basis of that contains the vectors and .

Q‒4: Let be a linear operator defined by .

[8 marks] Show that is a linear transformation.

[6 marks] Find a matrix such that for .

[6 marks] Describe the null space and the range of .

Q‒5: [20 marks] Find a formula for by diagonalzing the matrix.

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