Matrices!

Hello everyone :-)

All right let's begin with the nets and bolts (haha!) of matrices.
What is a matrix? Aside from its namesake movies that raked in millions, what is it?
It's a representation of numbers is what it is.
Disappointed? Don't be, matrices are one of the funnest and most important topics in mathematics. They crop up everywhere like tiny rabbits out of the vast ground that is maths.

Fine. Enough prosaic things for now.

How can you actually represent numbers? We've already discussed the number line, so that's a way. But the number line is limited that it is one-dimensional.
A 1-D, or one dimension, can represent one quantity. It could be simply numbers. Of dresses, chocolates, fruits, so on.

Now for 2-D.

You might be accustomed to the graphs you draw in classes, with the two axes- X and Y. What makes it 2 dimensional? The simplest answer is that there are two quantities. For example:
X could be the candies you want to buy and Y could be the money corresponding to the X candies you want to buy. For instance, 10 candies might cost 200 dollars (in a ridiculously fancy shop maybe!).

How do matrices help? With the matrix structure you can represent numbers in 2-D, and larger dimensions too!

For instance:

if X= [1 2 3 4 5 6]
Y=[20 40 60 80 90 100]

where X= no.of pens
Y= cost of the pens

You could represent it as:














Aside from it being so big( magnified so it will be clearer), what strikes you about this?
From a clumsy set of 2 data, which will be tiresome to write everytime you want to express it, you could draw a simple structure. The first row is X and the second row is Y.
Remember, matrices can be 1 dimensional too, like X and Y are both 1-D. But it can also be used in larger dimensions.

MATLAB IMPLEMENTATION:

Okay, as I discussed in my previous article, I will be providing snippets of matlab code.
Here it is. I've done a printscreen of the matlab windows- which consist of blank M-file and workspace.

Okay the code is as follows:

close all
clear all

X=[1 2 3 4 5 6];
Y=[20 40 60 80 90 100];

C=[X;Y];
disp(C)
plot(X,Y)

This I have written in an M-file, which is where you write your code in MATLAB.

The workspace window is on the left, where the output is displayed.

The plot function plots X vs. Y.

All right, that concludes it for today. As always, a pleasure and hope you learnt some interesting things! :-)

Never Too Late For Now!

Wow, that reminds me so much of the series 30 Rock!  :-)

Anyway, getting back to maths- I thought we could try matrices now. We're done with the number line and congruence, and I thought I might post some problems for today, but then this seemed urgent, particularly because I'm working on a concept called PCA (Principle Component Analysis) now, which is used heavily in imaging, face recognition and even, as I read in a recent paper, speech recognition! This is a fascinating concept that relies heavily on matrices. Very, very heavily. Also, this is a concept that is poorly explained in many of the tutorials I found. Now, I'm not going to be doing this immediately, but I hope to do so soon so more people out there find it's not such a mesh of complexity as they think it is!

Also, another thing- I'm going to be posting a few lines of code of MATLAB, an excellent, excellent piece of software that helps in computing in posts. A lot of technical papers rely on this, and I've started to do that too! Check these links to get to know this better:

http://en.wikipedia.org/wiki/MATLAB
http://www.mathworks.in/products/matlab/

Don't get scared- I'm going to go as slow as possible. I'm doing this because if I had come across MATLAB sooner, I wouldn't have gone through some of the rough patches I did.

However these are rules I strongly encourage you to follow:

1) Please do not start relying TOO heavily on this. You MUST be familiar with the concepts instead of asking it to do everything for you without you knowing what it's doing in the first place! Not following this rule may seem to be an easy way out but trust me, when you start writing codes with increasing complexity, you'll be stuck if you don't understand what each line does!

2)Try and learn concepts of C Programming as well. This will help you greatly in your projects as you get older. I'll try and start posting C lang. codes too as soon as possible!

3)Don't freak out. Don't be frightened. This is the most important rule. There is NOTHING you cannot do or understand. Give it time and ASK, ASK, ASK. Ask away if you have doubts. Remember, no judgment in brainstorming, understanding and asking valid doubts.

Okay, are we set? I will post a new entry and leave this one for people to get to know the rules and changes :-)

More Conditions To Prove Congruence

Hello again :-)
Right, so we were discussing conditions to prove congruence in the last post and covered two. Namely the SSS (Side Side Side) and the ASA (Angle Side Angle).
I'm going to repeat what I said before- that you need to apply intuition to many of these methods. Establishing congruence can be tricky- but if you look closely and apply both the conditions you're learning and a certain amount of intuition it is guaranteed that you can figure it out!
All right, now we discuss the three remaining methods.

3) SAS (Side Angle Side):
I like to call this the sister condition of the ASA. They are switched but similar.
To arrive at this condition and to understand it, first draw a side like we did in the ASA.
Assume AB=1 cm. (Fig. 1)
Please not it is not to scale in this blog- if I neglected to mention that earlier I'm sorry. I suggest that you  draw it to scale when you do it yourself, at least when you're still figuring out the conditions. Seeing is believing, they say!

Now, draw angle 30 degrees shooting out of A, and extend AC till AC=0.5 cm. You can change the numbers, but follow the steps. (Fig. 2)

Finally, draw a line to connect C and B. (Fig. 3)
The value of CB is not wholly important in this case, so I am not including the value.
If done right, it should look like the figures below.

Now do the same for DEF.
You should land up with:

Observe DEF and ABC are exactly the same. Hence ABC≅DEF

4) AAS: The Angle Angle Side

This condition is perhaps the most complex and you can understand this in a variety of ways. Let me show you my technique and I hope it helps!
Now, I want you to follow the following steps. Put away your pens and paper or keep them, but just try to imagine.
So we looked at the ASA. Let us try to use the ASA to understand the AAS.

We have a side AB= 1 cm.
Shoot an angle BAC= 30 degrees from point A.
Now we're left with line AC, but we really don't know what to do with it. How much should we extend it or retract it?
But we are given another piece of important information- the angle ACB.
I hope you're following this. This can be confusing, but it's learnt best slowly, steadily with lots of imagination. Put it this work now, and you can handle confusing sums later with a lot of confidence!

All right, getting back-
You may remember the property- sum of angles of a triangle is ALWAYS equal to 180 degrees.
When you have two angles, you can find out the third. In our case, we can find out angle CBA.
When we have angle CBA and angle BAC, along with side AB, what does that mean/

It means we're back to our good old ASA.

I'll leave the diagrams to you. Let me know in case of doubts!

5) Ah, finally. RHS. This is a special one- for right angled triangles ONLY!
This is the funnest one to prove, to me at least.
I assume you're familiar with Pythagoras's theorem:



where c= hypotenuse
a,b= sides of the right triangle

If you've forgotten, check this out.

So the RHS just states that if the hypotenuse of two right triangles are equal, and one of the other two sides are also correspondingly equal.

Look at the diagram. CB=EF (hypotenuse of triangle ABC= that of triangle DEF)

Also AC=DF
They are right triangles so they already have angle BAC= angle EDF
So, we have proved congruency. ABC≅DEF

You can do the same thing with sides AB and DE instead of AC and DF too.

Okay, so we're done for this session. Hope you've understood the concepts. Feel free to comment or e-mail :-)

The Congruence of Triangles

I have to apologize yet again for my delay in coming up with this post. There's been so many things to do, like work, work and coming up with even more excuses :-)
Anyway, let's not lose any more time and start.

So, in my last post, I had outlined the basics of congruence. Which means things being exactly the same.
Let's look at the usability of this concept. When you realize the importance of something, it's much more fun learning about it.

In many cases, we hear things like 'She looks SO MUCH like you!' or 'I have the EXACT same dress'. While this may be the truth, it may be an exaggeration as well. Looking like someone is not the same as being just like them- Anna and Melanie both might have blond hair, but Anna if Anna has green eyes and Melanie has grey eyes, the concept of congruence, of exactness, is immediately ruled out.
An exact same dress- if Anna has a brown shirt in a small size and Melanie has the same thing in a larger size, say medium, the dresses aren't exactly the same.

These may be small examples, but they are significant in learning about this concept that at many times proves frustratingly elusive.
But without this concept, we wouldn't have cola cans all being the same size, we might end up with a random size that takes all the fun out. We wouldn't have the impressive precision that mass produced products have- your favourite brand of shampoo, conditioner, chocolate, and so on.

This link talks about more examples- read it to get a further understanding.

All right, so I'm hoping you're starting to get the drift.
Congruency is exactness, fine, but how do you prove it?
A subjective way congruency can be expressed is saying something like 'I like the series 'Arrested Development' as much as I like 'Breaking Bad' '. As much as. In this case, equally.
Unfortunately, that's too arbitrary and people around you might disagree. So, to establish objectivity, we have a set of rules by which you can prove to everyone that two triangles (sorry, not series! not yet, at least!) are the same.

1)SSS: This stands for Side Side Side. Namely, if EACH side of one triangle is EQUAL to the CORRESPONDING side of the OTHER triangle, the triangles are said to be congruent.

Look at the triangles above. There's something missing. The naming of the triangle points which is VERY IMPORTANT TO ESTABLISH CONGRUENCY.

Better, I hope.
Now.
AB=DE
BC=EF
AC=DF


Yay! Thus ABC  DEF. Notice the difference from '='. It is merely the sign that represents mathematical congruence.

2)ASA: Angle Side Angle
I'm going to ask you to apply some intuition to this one. Imagine a triangle with  one of its sides AB. Imagine JUST the side. Assume it is of length 1 cm.

Now imagine two angles shooting out of its ends.
Extend the lines till they meet. Assume angles 30 degrees and 60 degrees.

OK, we have a triangle now!
Now do the same thing for line DE, again assuming DE= 1 cm etc. etc.. You land up with DEF. Try doing it yourself for practice.
You know ABC  DEF because we arrived at both the same way!

The remaining conditions will be discussed in the next post. Any questions, feel free to ask in comments or email :-)

Pointy Objects Like Triangles

I recently got to thinking about shapes. The curvy comfort of circles, and the calm, assured symmetry of squares and rectangles. The least comforting shape, though, was the triangle. Any way you hold it, you're not wholly comfortable. It pokes and shoves. Too many angles, students complain. Take the square- 90 degrees all the way. What's it with all these weird three sided things? They go this way and that- flat out and squat- and one obviously gets confused.

Save for all the complaints, the triangle is incredibly useful. In what?, you ask.
How about nachos? And the musical instrument, also called the triangle?
They are increasingly used in architecture, especially in bridges, and trigonometry is all about triangles, so, so incredibly important that even their peculiar shape shouldn't distract us.
So we'll talk more about this in today's post.

I assume you know how to differentiate between scalene, isosceles and equilateral types. If you have trouble with that, don't hesitate to ask, but for right now, we shall be moving on.
Let's talk about congruence. What is it, pray?
Congruence is nothing but the concept of things being the same shape and the same size. If there were two , lets go back to our famed doughnuts, chocolate doughnuts exactly the same size, and shape, they'd be called congruent in mathematical terms. Of course, we'd just call them delicious in real life though!
If there were two of you, like you and a clone of yours, wearing the same clothes, shoes, everything, you'd be called congruent, both of you.
Two objects are congruent if they are EXACT copies of one another. EXACT in bold. If your clone were wearing pink and you were wearing green, you wouldn't be congruent.

What we're primarily interested in now is the congruence of triangles.

Delving deeper into the number line

Hello again.
I can hear your groans, but please, I'll try to make it as interesting as I can.
Here's a picture of a..... doughnut?

Feeling better?
How about another one?

I'm actually hungry now. I'm going to take one away.

What happened here? Save for your scowls and hurt, what I did, can be expressed in mathematical terms:
I added a negative doughnut to the two doughnuts.

I added the additive inverse of 1 to 2.
2+(-1)
Think of the negative doughnut as a creamy, strawberry doughnut, different from the chocolate oozing one, but just as yummy (yummier? i think so!)

Okay then.
Next, what if I were to take away a 'negative' doughnut from the two former chocolate doughnuts, magically replaced?

2-(-1)

I subtract the additive inverse from the two doughnuts.
Imagine extracting a pink doughnut from thin air, imagine dragging it across the number line, watching it magically turn warm and chocolatey.
Voila, you have three doughnuts with you? Happy now? You should be.

Do not get confused. I'm trying to 'foodie' up the number line is all!

Envisage this:
-3-(-4)

Let's talk more about the doughnut analogy.
Take -3 in the number line. You need to subtract -4 from it, or add 4 to it. Both are THE SAME.

You have 3 STRAWBERRY doughnuts with you. (-3 is what you start with)
You must take away 4 STRAWBERRY doughnuts.( you need to subtract -4)

Take one away, you have 2 strawberry doughnuts.
Take another away, you have 1.
Take another away, you have, *sad face* 0.

What happens after this? We have one more step left.
Now is the interesting bit.
Think of the balance doughnut being dragged right of 0 in the number line.
You have one chocolate doughnut.

Confused? Try re-reading it for a bit.
Try this:

-7-(-12)?

You end up with +5, or 5 chocolate doughnuts. Think about it, and try to solve it on your own.

Remember, in your coursework, please refer to our chocolate doughnuts as positives, and the strawberry ones as negatives. The teacher wouldn't understand otherwise!

Somewhere over the.. number line

A few days ago, my friend came to me. Her sister, in the seventh grade, was confused with the number line concept, and would I explain it to her? I remember my days back when we were learning it. My teacher scribbling away at the board, and me, trying to catch up helplessly.

I did, eventually, but with difficulty.
So I thought I'd post about it, so it could help more confused souls out there.

The number line can be confusing. It's like an infinite lake, spreading in both directions like a confusing, endless thing. One misstep, and you think you've drowned. That is an exaggeration, but anything to liven this up, right? :-) I'm bad at jokes. 

Moving forward-

The start is easy enough. You have in one hand- a bunch of negative numbers, in the other a bunch of positive numbers and somewhere in between, maybe on your nose or forehead, a zero.

Imagine you're standing. On your left hand, imagine a magical ribbon shooting out. The longer the ribbon, the more negative the number. -10 would be a short one, -100 longer, -10,000 still longer, and negative infinity somewhere out there, endlessly moving forward.

Do the same on your right hand- out shoot the positives, smiley happy things shooting out.

Zero- bang in the middle.

Okay.

Please excuse me if you think I'm going too slowly. It's just that certain visualizations help a good deal. Remember- you'll be stuck with this seemingly innocent number line all your life- you wouldn't want to antagonize it. You'd be bickering for a l000ng time. See what i did there? ;-)

Now we know what the number line is, we've got an idea.

The next, irritating step is the movement. Add this, go right. Do this, slide left. Ah, the rights and lefts! The heres and theres!

Try to remember it like a dance- one of those fluid dances where the numbers are the glittery, fascinating dancers and the number line is the shiny, polished floor. Imagine you have a number, say 2. Add 3 to it. Where do you go from there? Left or right?

Think of 2 as a dancer. 3 is the number of steps she(or he, whatever pleases you), has to take.

Observe the sign. 3 is positive, so move the dancer '2' three steps to the right.

Is it confusing? I hope not. I merely referred to 2 the number as '2' the dancer.

Swish. Swish. Swish.

We reach 2+3=5.

What if the number isn't positive? Take -3.

Now the dancer '2' has to move three steps to the left. Another dance, perhaps a bit faster. They jump instead.

Jump. Jump. Jump.

2+(-3)= -1

here we are.

If you add a negative number to a negative number (in other words,  subtract a positive number from a negative one), what happens? can you visualize the dance?

-2 and -3:

-2+(-3)

The dancer '-2' moves three steps to the left.

-2-3=-5

Now, for an interesting topic-

The additive inverse.

Imagine you're standing in front of a mirror. You look into your image. It is you- it looks like you. But it is different. Imagine you're raising your left hand. Your image raises it's right hand.

What happened here? Lateral inversion in physics. You can think of it as an easy way to invert numbers.

The dancer '8' is looking at it's image '-8'.

The dancer '-3' is looking at it's image '3'.

Is it difficult to comprehend? I hope not. In doubt, think of yourself looking into the mirror. Think inversion.