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MATH6038: Week 5

February 27, 2014 in MATH6038 | 2 comments

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

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.

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 Mathematica. Mathematica 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.

MATH6038: Week 4

February 20, 2014 in MATH6038 | Leave a comment

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.

Manuals

Please bring E10.50 with you next week for your manual if you haven’t done so already. Those of you who didn’t get one last week will next week.

Week 4

We introduced matrices as linear maps and learned how to add, scalar multiply and multiply them. We introduced the idea of a matrix inverse and began talking about the Gauss-Jordan Algorithm for finding matrix inverses.

Week 5

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

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.32, Q. 1-7
  • p.37 exercises

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 Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Test

Provisionally I am looking at Week 7. Ye will get a sample test (format only) and notice two weeks in week 5.

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.

MATH6038: Week 2

February 6, 2014 in MATH6038 | Leave a comment

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.

Manuals

Please bring E10.50 with you next week for your manual if you haven’t done so already. Those of you who didn’t get one last week will next week.

Week 2

We looked at the solution space of linear systems.

Week 3

We will introduce matrices as linear maps. In Maple, we will learn how Maple handles linear systems with no and infinite solutions. We will also use the Maple Tutor to help us do the below exercises.

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 — on P.20 — at this point include (Q.2 should not be there):

  • Q. 1
  • Q. 4
  • Q. 5-7
  • Q. 8
  • Q. 9

Question 3

From P.20… we start by putting the system into augmented matrix form:

\left[\begin{array}{ c c c | c } 1 & 1 & -1 & 1 \\ 1 & 2 & -2 & 0 \\ -2 & 1 & 1 & 1 \end{array}\right]

r_2\rightarrow r_2-r_1,\,r_3+2r_1,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 3 & -1 & 3 \end{array}\right]

r_3\rightarrow r_3-3r_2,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 2 & 6 \end{array}\right]

r_3\rightarrow \frac12 \cdot r_3,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right]

r_2\rightarrow r_2+r_3,\,r_1\rightarrow r_1+r_3,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]

r_1\rightarrow r_1-r_1,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & 0 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]

\Rightarrow z=3,\,y=2\text{ and }x=2.

Maple Online & Wolfram Alpha

I am aware that some of us have not been able to install Maple… I have got to here:

Maple Download

If  you cannot get to this screen please email me and I might be able to help (sadly I don’t have admin rights on the computer I am working on right now so cannot find out do I need the 32 bit or the 64 bit…).

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 Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Test

Provisionally I am looking at Week 7. Ye will get a sample test (format only) and notice two weeks in advance.

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 solution set of a linear system.

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.

MATH6038: Week 1

January 30, 2014 in MATH6038 | Leave a comment

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.

Manuals

Please bring E10.50 with you next week for your manual. Those of you who didn’t get one last week will next week.

Independent Learning

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.

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 on the action link button next to the course 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, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

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

Week 1

After introductions, we looked at systems of linear equations and introduced the idea of writing them in augmented matrix form and simplifying using Gaussian Elimination.

Week 2

We will discuss how to determine if the linear system has a unique, infinite or no solution.

Test

Provisionally I am looking at Week 7. Ye will get a sample test (format only) and notice two weeks in advance.

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 visualising the solutions of linear systems.

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.

MATH6038: Week 5

March 4, 2011 in MATH6038 | 2 comments

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We looked at different types of data and how to present them. Notes here. When we look at histograms next week this section will be complete. In Maple we continued to boost our basic skills and looked at Maple’s LinearAlgebra package.

Next Week

I hope to finish looking at histograms. Then we will begin our study of probability, asking the question what is a random variable and when are random variables independent?

Reminder

The test will be held at 7.05 p.m. sharp on Wednesday 14 March.

Sample Test Answers

Here I give you links to how I checked answers quickly using Wolfram Alpha. It takes Mathematica code (they are the same company) and will probably decipher Maple code too (moreover Maple code with ‘adjustments’). Note that Wolfram Alpha gives us a lot more information than we need but that is the beauty of the thing really.

Question 1 (has a slight typo: should be -6z in the first equation)

Question 2

Question 3 — this implies that there are non-trivial solutions. Now that we know that it has non-trivial (non-zero solutions) so we can use Wolfram Alpha to solve for these (in terms of a parameter t.). Now let z=t so we have (x,y,z)=(t,t/2,t).

Bonus Questions

You could do without all the notation. The underlined would do.

1. Firstly the linear system can be written as a set of simultaneous equations:

a_{11}x_1+\cdots+a_{1n}x_n=b_1

a_{21}x_1+\cdots+a_{2n}x_n=b_2

\vdots+\cdots+\vdots=\vdots

a_{m1}x_1+a_{mn}x_n=b_m

The solution set is not changed by E_1 as this is simply writes the equations in a different order. Neither will E_2 change the solution (well if k\neq0) as multiplying one of the equations by a constant on both sides will not change the solution. Finally E_3 doesn’t change the solution set as we add the LHS_i (left-hand-side) of one equation to the LHS_j of another, while we add the constant b_i to b_j. However the LHS_i=b_i so we have just added the same thing to both sides and this does not change the solution.

2. A\mathbf{x}=\mathbf{b}\Rightarrow A^{-1}A\mathbf{x}=A^{-1}\mathbf{b}\Rightarrow I\mathbf{x}=A^{-1}\mathbf{b}\Rightarrow \mathbf{x}=A^{-1}\mathbf{b}.

3. You can solve the system for one rather than all variables.

MATH6038: Week 4

February 26, 2011 in MATH6038 | Leave a comment

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We said that a homogeneous linear system was one where all the constants are zero. We then proved the following:

Proposition

Let A\mathbf{x}=\mathbf{0} is a homogenous linear system of n equations in n unknowns where A is the coefficient matrix and the variables are x_1,\,x_2,\,\dots,\,x_n ( i.e. \mathbf{x}=(x_1\,\,x_2\,\,\cdots\,\,x_n)^T. Then the following holds:

  1. If \det A=0 then the system has (an infinite number of) non-trivial solutions.
  2. If \det A\neq =0 then the system has (a unique solution given by) the trivial solution (x_1=x_2=\cdots=x_n=0).

MATH6038: Sample Test

February 26, 2011 in MATH6038 | Leave a comment

Consider this notice for the test on Wednesday 14 March at 7 p.m (Wednesday fortnight).

Please find a sample test here.

MATH6038: Week 3

February 19, 2011 in MATH6038 | 2 comments

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We asked the question “when is a matrix invertible” and we answered the question by saying that a matrix is invertible when its determinant is non-zero. We didn’t give a definition of  a determinant but did see how to calculate it. Then we introduced Cramer’s Rule as yet another way of solving linear systems. We have a handout with some exercises on determinants and the chain rule.

We also did quite a lot of work on Maple in terms of the basics.

Exercise Solutions

I will give the solution to the first part of a question and then the answers to remaining parts. Some of code to write the matrices is very buggy so I’ve skipped some of them when they don’t compile straight away. For example, the answer below should have an extra line.

Read the rest of this entry »

MATH6038: Week 2

February 13, 2011 in MATH6038 | 3 comments

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We solved some more linear systems and began a study of matrices. Updated notes.

Exercise Solutions

I will give the solution to the first part of a question and then the answers to remaining parts.

Read the rest of this entry »

MATH6038: Week 1

February 3, 2011 in MATH6038 | Leave a comment

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 jippo@campus.ie and I will add you to the mailing list.

This Week

After introduction, we gave a motivation for studying linear systems and probability & statistics. We then looked at linear systems and how to solve them using elimination algorithms.

Typeset Notes

This is what we have thus far.

Maple Resource

A Maple tutorial we might look at in Week 3.