MATH6038: Week 7

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 next week (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 7

We continued talking about statistics including the standard deviation. We introduced frequency distributions. We skipped over the material that used the assumption of uniformity — we aren’t going to use it.

We introduced the assumed mean method and presented the formulae that are used to calculate the mean and standard deviation of a frequency class distribution using it. We also described the cumulative frequency distribution and how we can use it to find the median.

In Maple we revised for the test with the aid of Maple.

Week 8

We will have our test on Linear Algebra first and then after a break we will finish off the chapter on statistics and start talking about probability.

Independent Learning: Exercices

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 for your test
p.70 Q. 1-3
P.90 Q. 2

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 explaining why we have different formulae for the population and sample standard deviation.

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.

6 comments

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March 14, 2014 at 10:07 am

Student 52

Hi J.P.

I was wondering if you had an answer sheet to your sample paper that you attached to your mail. It’s just that I have it done and I was looking to see if the way I was doing them was right because in class we did a question like question 1 only it was with Gauss-Jordan not Gaussian Elimination.

Is there a difference in the answer because in part C you can do Gauss-Jordan to get $x$ and $y$ values.

Also Question 7 is the only difference between $CA^{-1}$ and $A^{-1}C$ …is the way you multiply the rows and columns?

March 14, 2014 at 10:18 am

J.P. McCarthy

Hi,

I don’t have an answer sheet but you send me your answers I can check them for you.

We did Q. 1 on P.18 of the notes using Gaussian Elimination. It is only part c that you could do by using the Gauss Jordan Algorithm to find the inverse of the matrix.

Big difference between $CA^{-1}$ and $A^{-1}C$ and the order in which you multiply matrices is important because, unlike with numbers, in general $XY\neq YX$ .

In fact, in Q. 7 it HAS to be $A^{-1}B$ . We show this here: suppose $AX=B$ and we want $X$ . We can undo the action of $A$ by multiplying ON THE LEFT on both sides with $A^{-1}$ :

$A^{-1}AX=A^{-1}B$

Now $A^{-1}$ undoes the action of $X$ so we just have $X=A^{-1}B$ .

Indeed not only is $X$ not equal to $BA^{-1}$ , $BA^{-1}$ isn’t even defined:

$\underbrace{B}_{3\times 1}\cdot \underbrace{A^{-1}}_{3\times 3}$ ,

can’t be done because the number of columns in $B$ does not match the number of rows in $A^{-1}$ .

Regards,
J.P.

March 14, 2014 at 2:04 pm

Student 53

How do you factorise $t^2+3t+2$ ?

March 14, 2014 at 2:12 pm

To be blunt this is something that you are supposed to be able to do before coming into this module.

You can find loads of information by googling ‘factoring quadratics’.

I did it in one line because I assumed ye could too but this is how I do it: if you can rewrite the middle term in terms of factors of the product of the first and last coefficients then you will be able to factor.

In this example, the first coefficient (of $t^2$ ) is one and the last is two. The product of these is two. The only factors of two is one and two and you can rewrite three — the middle coefficient — as two plus one; i.e. $3t=2t+1t=2t+t$ :

$t^2+3t+2=t^2+2t+t+2=t(t+2)+1(t+2)$ .

Note that we factored the first two terms and then we took something out of the second pair of terms such that $(t+2)$ would be left again.

Now $(t+2)$ is common to both $t(t+2)$ and $1(t+2)$ so we take it out as a common term to get

$t^2+3t+2=t(t+2)+1(t+2)=(t+2)(t+1).$

If this is all a little worrying at this point I have to point out that you won’t need to factor any quadratics in any MATH6038 Assessment.

For MATH6037 however, you will have to be an expert!

March 18, 2014 at 9:18 am

Student

Hi J.P.,

Just going over the sample test and I am stuck in few questions.

Q 1 I have strange results or I am doing something wrong

Q. 1 (a)
$\left[\begin{array}{cc|c}1 & 3 & 4 \\ 0 & 0 & 1\end{array}\right]$

Q. 1 (b)
$\left[\begin{array}{cc|c}1 & 2 & 3 \\ 0 & 0 & 0\end{array}\right]$

How do I describe the solution set $S$ ?

Q. 2 Why I need “ $t$ ” if I can get the values of

$x_1=-7,\,x_2=-24,\,x_3=3,\,x_4=2$ .

Regards,

March 18, 2014 at 9:21 am

Q. 1 (a) This looks good

The last row reads

$0x+0y=1$ ,

but

$0x+0y=0$

so this implies that we have no solutions (there is no $x$ and $y$ such that $0x+0y=1$ ).

Q. 1 (b) This also looks good.

Now we do have solutions. Now the difference between the number of variables $n=2$ ( $x$ & $y$ ) and the number of non-zero rows in reduced form $r=1$ is one so the number of parameters is $n-r=1$ . There is nothing saying that $y$ has to equal anything so we may let $y=t$ be any real number and then solve for $x$ .

We did Q. 1 on page 18.

Q. 2 First of all $x_4=8$ not $2$ …

Here we definitely have solutions. The difference between the number of variables $n=4$ ( $x_1$ , $x_2$ , $x_3$ and $x_4$ ) and the number of non-zero rows in reduced form is $r=3$ so the number of parameters is $n-r=1$ . The last row says that $x_4$ must equal $8$ . There is nothing that says that $x_3$ has to equal anything in particular so we may let $x_3=t$ be any real number and then solve for $x_2$ and $x_1$ .

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MATH6038: Week 7

Test

Maple Catch Up

Week 7

Week 8

Independent Learning: Exercices

Maple Online & Wolfram Alpha

Academic Learning Centre

Math.Stack Exchange

Additional Notes: E-Books

Recent Comments

Categories

J.P. McCarthy Maths

6 comments

Leave a Reply to J.P. McCarthy Cancel reply

MATH6038: Week 7

Test

Maple Catch Up

Week 7

Week 8

Independent Learning: Exercices

Maple Online & Wolfram Alpha

Academic Learning Centre

Math.Stack Exchange

Additional Notes: E-Books

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Recent Comments

Categories

J.P. McCarthy Maths

6 comments

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