I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test 1

Test 1 will be on Wednesday March 5 at 12:00 in B228 in Week 6. Please see the sample here. Everything in Chapters 1 & 2 is examinable but not ‘Round-Off Error and Partial Pivoting’.

Week 4

In the tutorials we looked at Gaussian Elimination. Also if you download Maple (see below), there is a Maple Tutor that is easy to use and will help you. Open up Maple and go to Tools -> Tutors ->Linear Algebra -> Gaussian Elimination.

In lectures we looked at what to do when a linear system has infinitely many solutions, as well as Cramer’s Rule and the iterative methods of Jacobi and Gauss-Siedel for approximating the solutions of diagonally dominant linear systems.

Week 5

In Monday’s tutorial we will look at the Jacobi and Gauss-Siedel Methods (as well as a bit of Cramer’s Rule).

In lectures we will look at what can happen when we use rounding when doing Gaussian Elimination.

Autumn 2013 Question 1 (b)

Carry out Gaussian Elimination on the following linear system until it is in reduced row form. Do not use decimal rounding.

\left(\begin{array}{ccc} 3 & -1 & 1 \\ 1 & 1 & -1 \\ 2 & -2 & 2\end{array}\right)\left(\begin{array}{c} x \\ y \\ z\end{array}\right)=\left(\begin{array}{c} 4 \\ 5 \\ -1\end{array}\right)

Show that there are an infinite number of solutions and find these solutions in terms of a parameter t.

Solution: We start by putting the system into augmented matrix form:

\left[\begin{array}{ c c c | c } 3 & -1 & 1 & 4 \\ 1 & 1 & -1 & 5 \\ 2 & -2 & 2 & -1 \end{array}\right]

Usually I would propose that we would do the ero r_1\rightarrow \frac13 r_1 here but we can be cuter by doing r_1\rightarrow r_1-2r_1 — or, even cuter again — r_1\leftrightarrow r_2,

\left[\begin{array}{ c c c | c } 1 & 1 & -1 & 5 \\ 3 & -1 & 1 & 4 \\ 2 & -2 & 2 & -1 \end{array}\right]

r_2\rightarrow r_2-3r_1 and r_3\rightarrow r_3-2r_1,

\left[\begin{array}{ c c c | c } 1 & 1 & -1 & 5 \\ 0 & -4 & 4 & -11 \\ 0 & -4 & 4 & -11 \end{array}\right]

r_2\rightarrow -\frac14 \cdot r_2, r_3\rightarrow r_3-r_2

\left[\begin{array}{ c c c | c } 1 & 1 & -1 & 5 \\ 0 & 1 & -1 & \frac{11}{4} \\ 0 & 0 & 0 & 0 \end{array}\right]

Now we are in reduced row form. There are solutions and n-r=3-2=1 parameter. We can let z=t where t\in\mathbb{R} be the parameter. Now the second row/equation reads

y-z=\frac{11}{4}\Rightarrow y=\frac{11}{4}+t.

The second row/equation reads

x+y-z=5\Rightarrow x=5-(\frac{11}{4}+t)+t=\frac94.

The solution set is given by:

\displaystyle \left\{\left(\frac94 , \frac{11}{4}+t,t\right)\,:\,t\in\mathbb{R}\right\}.

Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question which asks what are the applications of linear systems.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

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