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Australian National University

Student Number:

u

Mathematical Sciences Institute EXAMINATION: Semester 1 — End of Semester, 2020

MATH1013 — Advanced Mathematics and Applications 1

Book B — Algebra – with answer spaces

Exam Duration:      240 minutes.

Reading Time:           20 minutes.

Materials Permitted In The Exam Venue:

• Hand-written summary of any length, basic calculator (not a graphing calculator), unannotated paper dictionary.

Materials To Be Supplied To Students:

• Trig Cheat Sheet

Instructions To Students:

• The Algebra and Calculus sections are worth a total of 50 points each, with the value of each question as shown. It is recommended that you spend equal time on the Calculus and the Algebra papers.
• You have 20 minutes reading time then 4 hours to complete both papers.
• A good strategy is not to spend too much time on any question. Read them through first and attack them in the order that allows you to make the most progress.
• Your lecturer reserves the right to supplement this exam with an oral validation exam. The oral validation exam is an oral test to check that your level of understanding of the work on the exam is consistent with the exam mark.
• You must justify your answers. Please be neat.

Question 1                                                                                                                      10 pts

• Given the matrix A:

  1      0   2

                

A = −1    −2   1

  1      1   0

• Use row operations to find the inverse of A.                                      5 pts

• Use the inverse of A (or any other matrix method) to solve the system:     2 pts



                      x₁            +   2x₃          =          −1

  −x₁             −    2x₂  +     x₃           =          1

    x1

+    x₂                   =     0

Question 1 (continued)

• Use a determinant to find whether the following system

has a unique solution:                                                                                            3 pts



       x₁          +   x₂     +   x₃ =          0

  3x₁           −    x₃  =    −5

    x1

− x₂

+ x₃

=    −7

Question 2                                                                                                                      10 pts

• Consider a matrix A with REF:

−4   −1     0      1



                          

  0      4      9   −1

  0      0   −2     1

Justify your answers to the following questions:

• Are the columns of A linearly independent?                                               2 pts

• Does the system Ax = 0 have any non-trivial solutions?                            2 pts

• Do the columns of A span R³ ?                                                                     2 pts

• Is the matrix A invertible?                                                                              1 pt

Question 2 (continued)

• Consider the following statement about three vectors u, v, w in R³ :

If the set {u, v, w} is linearly independent, then the subset {u, v} is also linearly inde- pendent.

Either show that this statement is true, or provide a counterexample to show that it is false.                                                                                                                                 3 pts

Question 3                                                                                                                      10 pts

Consider the matrix A:





A = 



1   1   2    −1

2  2  4   −2







−1   0   4      3

• Find the Reduced Row Echelon Form of A.                                                            2 pts

• Find a basis for Nul A, the null space of A.                                               3 pts

• Find a basis for Col A, the column space of A.                                          2 pts

Question 3 (continued) Here again is the matrix A:





A = 



1   1   2    −1

2  2  4   −2







−1   0   4      3

• Use the name β for the basis for Col A found above.

  1

−  

Given u = −2 , find [u]_β , the coordinate vector of u with respect to β .        2 pts

  3

2

• Find v such that [v]

=   .                                                                                 1 pt

β       1

 

Question 4                                                                                                                      10 pts

A transformation T : R² → R³ has standard matrix:



          1   −2 

A =  −1      0   .

l



• Find T   

4

          3 −2 

1 pt



−3l

−5

    

• Find x such that T (x) = −3

3 pts

  1

• Determine whether T is onto and whether T is one-to-one.

Justify your answer.                                                                                               2 pts

Question 4 (continued )

• Recall the transformation T : R² → R³ with standard matrix:



          1   −2 

A =  −1      0   .

Find the range of T .

           3 −2 

You could express your answer in parametric-vector form, or as a Span.             4 pts

Question 5                                                                                                                      10 pts

• Express the following in standard form a + bi ,

(i)  (1 − 3i)²                                                                                                           1 pt

(ii)

1

2 − i

1 pt

(iii)

.            Σ 8

1

√2 (1 + i)

1 pt

• Express the following in polar form Rcis θ ,

1

(i)

2

2cis ^π

1 pt

(ii) ( ¹ cis^π )(2cis^π )²                                                                                                1 pt

4        6               6

Question 5 (continued )

. π Σ

. 2π Σ

• (i) z₁ = cis

3

and z₂ = cis

are two roots of a quartic polynomial P (z) with

3

real coe@cients and leading coe@cient equal to 1.

Find P (z) in expanded form P (z) = z⁴ + bz³ + cz² + dz + e .                        4 pts

(ii) Another polynomial H (z) , of degree 6, also has real coe@cients with leading coe@cient equal to 1 and four of its six roots are:

, z

z1

= cis . π Σ

3

= cis . 2π Σ

, z

2                  3

₃ = cis(0) , z₄

= cis(π ) .

Find H (z) in expanded form.

(Justification is optional, you can just write it down).                                   1 pt

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