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.

Pending…

## Week 6

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

In lectures we looked at what can happen when we use rounding when doing Gaussian Elimination. Then we started Laplace Methods.

## Week 7

In Monday’s tutorial we will look at doing some partial fractions. I am going to rob and adapt these exercises.

In lectures we will start looking properly at the Inverse Laplace Transform.

## 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, this and this question are looking for but not finding intuitive ways to think about the Laplace transform.

## 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

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.

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.

## Week 6

We finished our of vectors by looking at the application to work and moments. We also did a review of MATH6015 differentiation.

## Week 7

We will study curves more complicated than those of the graph of a function: namely parametric curves and implicit curves.

## Test 2

The second test will be on in Week 11. That is Tuesday 8 April February at 11:00 in B214. Ye have the test that I gave last year in the notes.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

## 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 about understanding the dot and cross product.

## 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

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 ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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

LATE EDIT: I just realised that I included no questions like Q.1 on P.32 of the notes… they are also examinable (matrix arithmetic). Also Q. 7 should read “Verify your answer for $z$ using Cramer’s Rule

The test will be from 7.05-8 pm in Week 8 (Wednesday 19 March). I won’t give a sample but instead look here for a selection of test & exam questions. You  can find a summary of Chapter 1 methods here.

Also I explained that some questions involving finding the inverse of 3×3 matrices using the Gauss-Jordan Algorithm can end up very messy in terms of the numbers.  Questions like P. 44 Q.1-5 are fine as are P. 52 Q. 2, 3 (iv)-(vi)

Stay away from questions like the examples 3 & 4 on P. 49 and P.52 Q. 3 (iii), (vii)-(x) and P.62 Q.2, 3. — the numbers here are just too messy.

## Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on.

## Week 6

We solved a number of linear systems using the  inverse matrix method. Note however that this cannot work if the solution is not unique… we used a property of matrices called the determinant to decide in advance if a certain class of linear system, homogenous systems, had a unique solution or not. If the matrix (of coefficient) is invertible then the solution has to be unique and we saw that the matrix (of coefficients) is invertible if and only if the determinant is non-zero. Finally we will looked at Cramer’s Rule: a method for solving for only one of the variables of a linear system. We then began talking about statistics and got as far as discussing the difference between the mean, the median and the mode.

## Week 7

We will go as far into the chapter on statistics as possible. In Maple I will allow you to revise for the test with the aid of Maple. I will write a worksheet which includes the sample questions. Remember Maple runs from 6-7.10 and 8.50-10. The theory class runs from 7.15-8.45.

## Independent Learning: Exerices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

• the sample questions
• p.52 Q. 3 (iv)-(vi)
• P.62 Q. 1 and 4
• P. 68 Q. 1-4

### 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

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.

If you have any problem with this take a print screen and send it to me and I will try and sort you out.

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!

## Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. 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 about the determinant.

## 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 ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

I went through the first four weird facts from this talk with the some of the fifth and sixth class pupils at Gaelscoil an Ghoirt Alainn, Mayfield.

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.

## Homework

Will be discussed in class on Monday.

## Week 9

We finished our study of the Doubling Mapping and started talking about complex numbers and how they arise both algebraically and geometrically.

## Week 10

More on complex numbers.

## Exercises

For the Week 10 tutorial you should look at Q.44, 45, 52, 53, 56-60. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## 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 conjecture about solving a complex numbers problem geometrically.

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 5

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

In lectures we looked at what can happen when we use rounding when doing Gaussian Elimination… the answer is we have to do partial pivoting. We also began the section on the powerful theory of the Laplace Transform.

## Week 6

In Monday’s tutorial we should be looking at our  sample paper — you should print off a copy for yourself. Otherwise look at the sample in the notes but replace the partial pivoting question by a Jacobi/Gauss-Siedel question.

In lectures we will look at partial fractions — a vital technique in the use of the Laplace Transform in solving differential equations.

## 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 answer which says that partial pivoting can actually be really bad in theory (give inaccurate solutions), but almost always works well.

## 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

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.

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.

## Week 5

We continued working with the dot product and then introduced the cross product.

## Week 6

We will study the application of vectors to work and moments.

## Test 2

The second test will be on in Week 11. That is Tuesday 8 April February at 11:00 in B214. Ye have the test that I gave last year in the notes.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

## 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 about generalising the vector product to more than three dimensions.

## 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

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 ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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

Due to the lost night in the storm, I am now putting the test in Week 8. Ye will get a sample test (format only) in week 6. It will be on Chapter 1.

## Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on.

## Week 5

We found some matrix inverses using the Gauss-Jordan Algorithm and saw how matrix inverses are useful in solving matrix equations. In particular, we saw that we can rewrite a linear system as a matrix equation.  In Maple, we saw how Maple handles linear systems with no and infinite solutions. We also used the Maple Tutor to help us do some matrix manipulations.

## Week 6

We will solve a number of linear systems using this inverse matrix method. Note however that this cannot work if the solution is not unique… we will use a property of matrices called the determinant to decide in advance if a certain class of linear system, homogenous systems, have a unique solution or not. If the matrix (of coefficient) is invertible then the solution has to be unique and we will see that the matrix (of coefficients) is invertible if and only if the determinant is non-zero. Finally we will look at Cramer’s Rule: a method for solving for only one of the variables of a linear system.

## Independent Learning: Exerices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

• p.44 exercises
• p.46 exercises
• p. 52 Q. 1, 3(i),(ii) & 4

### 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

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.

If you have any problem with this take a print screen and send it to me and I will try and sort you out.

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!

## Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. 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 about matrix inverses.

## 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 ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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.

## MCQ Results

As discussed previously, the MCQ results will come out the day you hand in the homework: 11 April 2014. I will send the homework next week.

## Week 8

We finished describing Chaos Theory and began proving that the Doubling Mapping is chaotic.

## Week 9

We will finish our study of the Doubling Mapping and possibly start talking about complex numbers.

## Exercises

I have emailed ye a copy of the exercises. For the Week 9 tutorial you should look at Q.46 and Q. 47. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## 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 conjecture about the iterates of the tangent function.

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

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.