Faculty of Computer Studies MT132 – Linear Algebra Take Home Exam for Final Assignment (Spring 2020/2021) Contents Warnings and Declaration………………………………………………………………            1 Question 1

Faculty of Computer Studies

MT132 – Linear Algebra

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

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.

Answer the following questions:

Q?1:

1. [12 marks] Use the inverse of a matrix to solve the linear system:

x1-x2-x3=5x2+x3=-1x1-2x2-x3=2

.

2. [8 marks] Let a1a2a3b1b2b3c1c2c3=-4

   

   . Find 2a32a22a1b3-a3b2-a2b1-a1c3+3b3c2+3b2c1+3b1.

   

Q?2:

1. [8 marks] Determine whether S={1,0,-1,2,1,0,3,1,-1,1,1,1}

   

    spans R3

   

   .
2. [6 marks] Find all c∈R

   

    for which S={c2,0,1,0,c,0,1,2,1}

   

    is a linearly independent set of vectors in R3

   

   .
3. [6 marks] Determine whether W=x,y,z∈R3 x2+y2=z2

   

    is a subspace of R3

   

   .

Q?3:

1. [6+4 marks] Let W=span{(1, 1, 0), (1, 0, 1), (2, 1, 1), (0, 1,-1)}

   

   .

1. Is (9,0,9)

   

    in W

   

   ? Justify your answer.
2. Is W=R3

   

   ? Justify your answer.

2. [6+2+2 marks] Let A=1123113300100120

   

   .

1. Find the reduced row echelon form of A

   

   .
2. Find a basis for the subspace spanned by the rows of A

   

   .
3. Find a basis for the subspace spanned by the columns of A

   

   .

 Q?4: Let T:R3?R3

 be a linear operator defined by  Txyz=x-y-z2z

.

1. [8 marks] Show that T

   

   is a linear transformation.
2. [6 marks] Describe R(T)

   

   . What is the dimension of R(T)

   

   ?
3. [6 marks] Find a basis for the null space of T

   

   .

Q?5: Let A =000014023.

1. [8 marks] Find the eigenvalues of A

   

   .
2. [12 marks] Find a nonsingular matrix P

   

    and a diagonal matrix D

   

    such that D=P-1AP

   

   .