M132: LINEAR ALGEBRA

M132: LINEAR ALGEBRA
Tutor Marked Assignment

Cut-Off Date: Week of December __, 2016 Total Marks: 60



Contents Page
Feedback form ……….……………..…………..…………………….…...….. 2
Question 1……………………..………………………………………..……… 3
Question 2……………………………..………………..……………………… 4
Question 3………………………………..………………..…………………… 4
Question 4………………..……………………………………..……………… 5
Question 5 ……………………..………………………………………..……… 5
Question 6 ……………………………..………………..……………………… 6


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M132 TMA Feedback Form

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QUESTION 1 2 3 4 5 6
MARK 10 10 10 10 10 10
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Tutor’s Comments:


The TMA covers chapters 1 and 2. It consists of 6 questions, each worth 10 marks for a total of 60 marks. Solve each question in the space provided. You should give the details of your solutions and not just the final results.
Q−1: [5×2 marks]
Answer each of the following as True or False (justify your answer):

a) If X1 and X2 are solutions of the equation AX = B (B ≠ 0), then X1 + X2 is also a solution.




b) If A and B are 2 x 2 matrices, then the sum of the terms on the main diagonal of AB – BA is zero.



c) Suppose A is a 2 x 2 matrix. If A is invertible, then At is also invertible.




c) Suppose that S = {v1, v2, v3} is a linearly independent set in Rn; then T = {v1, v2, v1 + v2 + v3} is also linearly independent.





e) For





Q−2: [7+3 marks]
Let
a) Find A-1.
b) Solve the linear system AX = B, where








Q−3: [4+2+4 marks] Consider the linear system Ax = b where

a) Solve the linear system;
b) Give a particular solution Sp;
c) Solve the homogeneous equation Ax = 0.

Q¬−4: [4+4+2 marks] Let

a) Find a matrix B in reduced-row echelon form that is row equivalent to A;
b) Find det(A);
c) Calculate


















Q¬−5: [7+3 marks] Consider the linear system ,
a) Solve the linear system by row-reducing the corresponding augmented matrix;
b) Find the values of h for which the following set of vectors is linearly independent: .



Q−6: [2+4+3 marks]: Given the vectors

a) Show that the set is linearly dependent;
b) Determine whether the set is linearly dependent;
c) Find the scalars such that v4 can be written as ;
d) Find all vectors that can be written as a linear combination of the set .