A Hello from Me, Linear Systems and Measure for Measure

Introductions

HELLO! THIS IS MY BLOG!

HERE IS A PHOTO OF ME LOOKING HAPPY WITH MYSELF FOR CREATING THIS BLOG!

I've tried making blogs before and blogged about my thoughts and crazy dreams but I thought that I would do something different. So here are my experiences as an amateur actress, a LARPer and a mechanical engineering student. I'm going to be using this as a bit of a study resource as well as a thought map so if you don't like learning things, best not read the sections with titles like "algebra rules for multivariable control systems" or "thermoplastics processing". So i'll get on with it :D

Another Play!

FANTASTIC news today! I've been shortlisted for a part in Measure for Measure (a play from the wonderful Shakespeare) at shoreside theatre. Recalls are next Tuesday and needless to say I am very very excited. The last time I was in a play was last year in December as a pantomime villain Dan Dini at Mairangi Players (picture top right) and the last time I was in the Shoreside Summer Shakespeare was when I was 15. I played William the simpleton in As You Like It (picture bottom right).

Measure for measure is actually a really interesting play about morality and different views on *cough* fornication. I'm currently making my way through the play with reference to the English interpretation (because Shakespeare is totally a foreign language right? :P ). Here's the link to the text with modern interpretation http://books.google.co.nz/books?id=Vg8wXH-2kvoC&printsec=frontcover#v=onepage&q&f=false

Jordan and Amy and Mark and Jackie have also been called back so needless to say I'm really really excited for Tuesday and the rest of the play!!

Linear Independence

Firstly, here's an introduction to matrices if you have no prior knowledge but really want to challenge yourself by reading on. http://www.purplemath.com/modules/matrices.htm

Basic Concept 

So another exciting thing that happened today is that I understand linear independence! Linear independence is a relationship between vectors (in control systems, these vectors are mostly within matrices). The basic concept is that if two vectors are linearly dependent they can be multiplied by a constant and summed together to cancel each other out. 

Visualization of Concept

A good way to visualize a vector is as a coordinate or an arrow. Here's a picture I drew for you!

Linearly Dependent Arrows

The red and the orange arrow are linearly dependent. Visually we can see this as they are both on the same straight line with the same absolute gradient (somewhat limited by my drawing skills in paint but you get the point). The orange vector can be multiplied by a constant of approximately 2 and added to the red vector to cancel out completely. A mathematical example of linear dependent vectors is [2,4] and [4,8] because 2*[2,4] + -1*[4,8] = [0,0]

Linearly Independent Arrows

My second drawing is of linearly independent vectors. They have different gradients so no matter how much you make the arrows grow or shrink (mathematically speaking, no matter what constants you multiply them by) they will never be able to be summed in order to cancel each other out. A mathematical example of linear independent vectors is [2,4] and [3,1]. the ratio of the first and second numbers is different in each of the vectors so it is impossible to times them by a constant in order for them to cancel each other out. No, you are not allowed to multiply them by zero in order to cancel them you sneaky sneaky thing. Also vectors do not count as constants so don't even think about multiplying one of the vectors by another vector.

Rank Limitations and Magic

The magical thing about vectors (and this is really awesome) is that when you have a matrix, you can never ever ever ever have a matrix with more linearly independent vectors than the minimum dimension of the matrix.

For example: if you have a 4x2 matrix [3,2;  4,5;  6,2:  3,1] you can never have more than 2 linearly independent vectors. Any conceivable vector can be created by forming a linear equation using one or more of the linearly independent vectors multiplied by constants. e.g [3,2] = a[4,5] + b[6,2]. We know it is possible to find out a and b because there are two constants and two unknowns!!! MAGICAL RIGHT??

Determining Rank (Full Rank?)

Determinant Method of Determining Rank

There are two nifty ways to figure out if a matrix is full rank. The first method is to find the determinant of the matrix and if it is not equal to zero, the matrix is full rank (det(A) /= 0 ---> A is full rank) This is best used on equations with an equal number of rows and columns. You can use it on 2xn matrices by going through and testing each column against the other columns for linear dependence. This is illustrated to the left. All combinations of vectors need not be tested. Some vectors can have an implied linear dependence. For example if the vectors in the green square have a determinant = 0 and the vectors in the blue square have a determinant = 0, it is implied that vector [a,b] is linearly dependent to vector [d,e] even though this vector combination was not formally tested. Cool right? 

(or not)

The second method of testing is more suited to matrices with unknowns and such or long 2xm matrices if you can't be bothered finding out ALL the determinants. You can simply look at individual vectors, and figure out if they can be multiplied by a constant to create another set of vectors within the matrix (this is only for the case where a vector is dependent on only one linear dependent vector but in trickier cases a vector can be dependent on two or more). If this is possible then only one of the vectors is counted as linearly independent. All the other vectors are clones if you will. They may have different hair cuts and clothing but each has the same DNA. By using this methodology and using implied dependence techniques a solution can be very quickly reached. Here is an example of a problem suited to this second method of doing things:

Topic For Business Plan 

In 703 we have to find an idea for an innovation to market and I have decided to do something a bit larpy. I want to make a large scale GM app for WOD games. Something that will hopefully make large scale combats a lot quicker and smoother to add to the immersion of the game. It allows players to be alerted when their initiative is reached and sends the success of their rolls directly to the GMs. It can be downloaded onto smart phones and bought from the WOD creators for a small fee. The plan is that it will not only make a profit (about 50c per download) but it will also help grow the game which is somewhat limited by how awkward combats can be. I'm pretty excited about this idea so LARPers watch out! I will be doing market research on you!!! :D

Goodbye!

I doubt anyone will read to the end but congrats if you have! If I am good I will post again tomorrow. Otherwise you may abuse me for clearly not doing my homework! CYA!