This document provides instructions for a 45 minute MATLAB test with 6 questions worth a total of 25 marks. It instructs students to start MATLAB, load the test data, and provides some MATLAB commands that may be useful. It also notes that answers can be exact or numerical to 4 significant figures or decimal places.

1. MAST10007 Linear Algebra
MATLAB Test
Test duration: 45 minutes
This paper has 6 test pages
Please complete all the following details.
Name:
.....................................................
Student Number:
.....................................................
Tutor’s Name:
.....................................................
s
Tutorial Time and Day:
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Instructions to Students:
This examination is designed to test your ability to calculate efficiently with the aid of MATLAB
as well as your knowledge of the material covered so far in the subject. Some questions test your
understanding of the material covered in lectures, and do not necessarily require MATLAB. No partial
credit is given, so please carefully check anything typed into MATLAB, and check the output of any
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programs used.
There are 6 questions. The number of marks for each question is indicated and the total is 25.
(1) Before starting the test complete the following steps:
Start up MATLAB. (This creates the folder Student Data:MATLAB.)
Copy the folder TEST
from Maths & Stats - Lab MaterialMAST10007 (the tree)
to Student Data:MATLAB (the star).
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Set the path in MATLAB to include Student Data:MATLAB with subfolders.
Run the command 00SAMPLE to load the test data into your MATLAB session.
(2) Some MATLAB commands:
rref(A) gives the reduced row echelon form of A
A’ is the transpose of the matrix A
det(A) gives the determinant of the matrix A
eye(n) gives the identity matrix of size n × n
rank(A) gives the rank of the matrix A
mod(x,2) gives x modulo 2
sqrt(x) gives the square root of x
(3) Answers may be either exact or a numerical solution correct to 4 significant figures or correct to
4 decimal places.
Do not turn over the page until instructed to do so.

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3. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 1. Verify that the following matrices are already loaded in your MATLAB session (if not, repeat
step 1 in the instructions):
18 −50 −15
 
1 6 8 22 45

 0 1 2 6 5 −12 −6 13 


 1 6 8 21 17 −49 −14 42 

 0 3 7 24 21 −44 −25 55 
A= 

 −3 −12 −5 5 11 29 −33 34 


 0 1 −2 −15 −16 17 19 −42 

 0 −3 −5 −12 −9 28 11 −23 
−3 −4 3 20 18 −18 −40 55
12 11 −12 −5 1 −12 −6
 
10

 −10 11 9 −5 −3 −12 −9 1 

−10 −4 6 −9 −11 −7 6 −10 
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
 
 −3 1 1 9 9 −3 −1 0 
B= .

 5 5 −13 −4 −3 −2 −9 5 

 −4 12 7 6 −10 7 −5 2 
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 
 −10 −9 −12 −12 9 −2 −2 12 
−4 1 5 4 12 −6 1 −5
(a) What is the entry in the 3rd row and 1st column of BA?
Answer: 59
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(b) What is the rank of A?
Answer: 4
(c) What is the determinant of B + 7I where I is the 8 × 8 identity matrix?
Answer: -73054752
(d) Let u be given by the 8th row of A and v be given by the 5th column of B. What is the value
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of the dot product u · v?
Answer: 582
Mathematics and Statistics 1 of 6 University of Melbourne

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5. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 2. Verify that the augmented matrix C of the following system of linear equations is already loaded
in your MATLAB session (if not, repeat step 1 in the instructions):
− x4 + 3x5 + 4x6 + 3x7 = −3
−x1 + x2 − x3 + 3x4 − x5 + x6 − 4x7 = 9
−x1 + 2x2 − x3 + 3x4 − x5 − x6 − 4x7 = 7
x1 − 2x4 + 2x5 + x6 + 3x7 = −6
− x2 + x3 + x4 − 6x5 − 8x6 − 4x7 = 3
2x2 − 2x3 + 2x4 + 3x5 + 6x6 − x7 = 6
(a) Write down the reduced row echelon form of the matrix C.
Answer:
 
1 0 0 0 0 1 1 0
s

 0 1 0 0 0 −2 0 −2 

 0 0 1 0 0 0 2 −2 
rref(C) =  .

 0 0 0 1 0 2 0 3 

0 0 0 0 1 2 1 0 
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
0 0 0 0 0 0 0 0
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(b) Find the general solution of the system of linear equations.
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Answer:
(x1 , x2 , x3 , x4 , x5 , x6 , x7 ) = (0, −2, −2, 3, 0, 0, 0)+s (−1, 2, 0, −2, −2, 1, 0)+t (−1, 0, −2, 0, −1, 0, 1)
Mathematics and Statistics 2 of 6 University of Melbourne

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7. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 3. Verify that the following matrix is already loaded in your MATLAB session (if not, repeat step
1 in the instructions):
−2 −8 9 −20 −11 −6 −15
 
2

 2 −3 −15 16 −27 −26 1 −13 


 −3 1 6 −10 6 23 −3 −12 

D=
 2 −3 −12 17 −12 −36 10 15  .


 2 −5 −21 25 −29 −47 11 2 

 −3 6 25 −30 39 53 −8 5 
−1 0 −2 1 1 −4 6 3
Let S be the following set of vectors:
 
−2 −8 −20 −11 −6 −15
             

 2 9 

 2   −3   −15
   16   −27   −26   1   −13 


               

s
 −3   1  

 6   −10   6   23   −3   −12 

              
S =  2  ,  −3  ,  −12 , 17 , −12 , −36 , 10 , 15  .
  
 2   −5   −21
            

    
 
  25  
  −29  
  −47  
  11  
  2 


 −3   6   25

−30 39 53 −8 5

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           

 
−1 −2 −4
 
0 1 1 6 3
(a) Is the set S linearly independent ?
Answer: No
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(b) Is S a spanning set for R7 ?
Answer: No
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(c) Find a subset {v 1 , v 2 , v 3 , v 4 , v 5 } ⊆ S (ie., a subset having five elements) with the property that
{v 1 , v 2 , v 3 , v 4 , v 5 } is linearly independent.
(you may indicate the appropriate subset by circling vectors).
Answer: Vectors listed 1,2,3,4,5, i.e. the subset
 
−2 −8 −20
       

 2 9 

 2   −3   −15
   16   −27 


         

 −3   1  

 6   −10   6


        
 2  ,  −3  ,  −12 , 17 , −12  .
  
 2   −5   −21
      

    
 
  25  
  −29 


 −3   6   25

   −30   39 


 
−1 −2
 
0 1 1
Mathematics and Statistics 3 of 6 University of Melbourne

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9. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 4. For this question you will be working in mod 2 arithmetic, that is with scalars Z2 .
Let  
1 0 0 0 1 1 1 1
 0 1 0 0 1 0 1 1 
E=  0 1
.
0 1 1 1 0 1 
1 1 0 1 0 1 0 1
The matrix E should already be loaded in your MATLAB session, as should a matrix F having as
columns the vectors v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ∈ Z8 . (If that is not the case, repeat step 1 in the instructions.)
2
Circle YES if the vector is in the solution space of E. Otherwise circle NO.
v 1 = (0, 1, 0, 0, 0, 1, 1, 0) YES NO
v 2 = (0, 0, 1, 0, 0, 0, 0, 0) YES NO
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v 3 = (1, 0, 0, 0, 1, 1, 1, 0) YES NO
v 4 = (0, 0, 1, 1, 0, 1, 1, 1) YES NO
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v 5 = (0, 1, 1, 0, 0, 1, 0, 1) YES NO
v 6 = (1, 1, 1, 0, 1, 1, 0, 1) YES NO
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Mathematics and Statistics 4 of 6 University of Melbourne

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11. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 5. Verify that the vectors b1 , . . . , b5 listed below are already loaded in your MATLAB session (if
not, repeat step 1 in the instructions).
Consider the basis of R5 given by
B = {b1 , b2 , b3 , b4 , b5 }
with b1 = (3, 4, 4, 2, 2)
b2 = (1, 1, 0, 0, 1)
b3 = (−4, −4, −3, −1, −3)
b4 = (−4, −6, −6, −3, −3)
b5 = (−5, −2, 0, 2, −3)
Let S denote the standard basis for R5 .
(a) Form the transition matrix P = PB,S and use it to calculate the coordinate matrix of the vector
v = (−2, 5, 5, −5, −2) with respect to B. (There is no need to write down P on the answer
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paper.)
 
−25
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
 9 

[v]B = 
 33 

 −34 
−12
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(b) A linear transformation T : R5 −→ R5 is given by:
T (x1 , x2 , x3 , x4 , x5 ) = (x1 + x2 + 5x3 + x4 − 2x5 , x2 − 3x3 + 5x4 + 3x5 , x3 − 2x4 , x4 + 3x5 , x5 )
Write down the matrix representation of T with respect to S.
 
1 1 5 1 −2

 0 1 −3 5 3 

[T ]S =  0 0 1 −2 0 
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 
 0 0 0 1 3 
0 0 0 0 1
(c) Find the matrix representation [T ]B of T with respect to B. Write down the (1, 2) and (5, 3)
entries of [T ]B .
[T ]B (1, 2) = 6 [T ]B (5, 3) = −4
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13. MAST10007 Linear Algebra TEST 00 Semester 1, 2012
Q 6. Verify that the matrix G given below is already loaded in your MATLAB session (if not, repeat
step 1 in the instructions).
The following defines an inner product on R5 :
 
y1
y2 
 
(x1 , x2 , x3 , x4 , x5 ), (y1 , y2 , y3 , y4 , y5 ) = x1 x2 x3 x4 x5 G y3 
 
y4 
y5
where  
1 4 −2 −2 1

 4 25 −11 7 10 

G=
 −2 −11 14 −1 5 .

 −2 7 −1 30 10 
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1 10 5 10 27
Let
u = (48, 1, −1, −3, −5) and v = (0, −1, 0, −2, 3) .
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Using the inner product specified above
(a) find the inner product of u and v:
u, v = −37
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(b) find the length of u:
u = 64.6838
(c) find the length of v:
v = 15.3623
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(d) find the cosine of the angle θ between u and v:
cos(θ) = −0.0372349
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