Solving A System of Linear Equations

“Your proof is wrong. But maybe you are Batman and the statement actually is true.” - from this

   Two linear systems are called equivalent if they have the same solution sets. Our approach to solving a system of linear equations is to transform the given system into an equivalent one that is easier to solve.

Solve the system:

Solution:
   Starting from the last equation and working backward, we find successively that z = 2, y = 5 - 3(2) = - 1, and x = 2 + ( -1) + 2 = 3. The procedure used to the above system is called back-substitution method. We now turn to the general strategy for transforming a given system into an equivalent one that can be solved easily by back substitution.

Solve the system:

Solution:

   We first need to eliminate the variable x from equations 2 and 3. Observe that subtracting appropriate multiples of equation 1 from equations 2 and 3 will do the trick. Next, observe that we are operating on the coefficients, not on the variables, so we can save ourselves some writing if we record the coefficients and constant terms in the matrix where the first three columns contain the coefficients of the variables in order, the final column contains the constant terms, and the vertical bar serves to remind us of the equal signs in the equations. This matrix is called the augmented matrix of the system. 

  There are various ways to convert the given system into one with the triangular pattern we are after. The steps we will use here are closest in spirit to the more general method described in the next section.We will perform the sequence of operations on the given system and simultaneously on the corresponding augmented matrix. We begin by eliminating x from equations 2 and 3.

   This is the same system that we solved using back substitution, where we found that the solution was [3, 1, 2]. This is therefore also the solution to the system given in this example.Why? The calculations above show that any solution of the given system is also a solution of the final one. But since the steps we just performed are reversible, we could recover the original system, starting with the final system. (How?) So any solution of the final system is also a solution of the given one. Thus, the systems are equivalent (as are all of the ones obtained in the intermediate steps above). Moreover, we might just as well work with matrices instead of equations, since it is a simple matter to reinsert the variables before proceeding with the back substitution. 

DISCLAIMER

   This post is directly copy-pasted from the book "Linear Algebra, A Modern Introduction" by David Poole

Not really but, OMG, I finally understand it. Thank you, Ma'am Nicole and Regional Science High School III, for answering this question of existential dread, and for being the building blocks of knowledge and wisdom for what I am right now as I type and post this.

Parameters

“I mean that gods do not limit men. Men limit men.”  

― Tom Robbins, Jitterbug Perfume

   This, by part, is true. Gods do not forbid the endless meaning of life, by which, men insists that the possible is out of our reach.

   In my lesson for today, I will expand the word "parameters" in the world of Linear Algebra

   They define parameters (in Statistics) as a value, usually unknown (which, therefore has to be estimated), used to represent a certain population characteristic. In Linear Algebra, people define parameters as to represent the infinite number of solutions of the equation; it is convenient to introduce a 3rd variable t.
That is just the definition. But what is the purpose of a parameter? It is to limit the infinite solution. For example:
     

• x1 + 2x2 = 4     [You transpose 2x2 to 4 which gives,]
• x1 = 4 - 2x2      [You cannot multiply the equation by 1/4 on both sides, it will give x1 a fraction. What to do? You set out a paramater]
• x2 = t, where t is a whole number; real number; counting number, etc.

   We therefore conclude that x1 = 4 - 2t

   To fully understand on what number to set in the parameter, you have to know what the nature of numbers are and their definitions:

• Counting number - 1 to positive infinity
• Whole number - all numbers of denominator of 1 
• Integers - negative infinity to positive infinity, excluding 0
• Real numbers - numbers that are not imaginary
• Rational numbers - can be expressed in fractional numbers
• Irrational numbers - numbers that are non-terminating and non-repeating

Another example:

• 3x + 2y - z = 3   [Transpose 2y - z to the right to get x]
• 3x = 3 - 2y +z    [This is where you set a limit]
• Let: y = s; z = t where s and  t  are real numbers
• 3x = 3 - 2s + t   [The leading coefficient must be 1, so we multiply the equation by 1/3]
• 3x/3 = 3 - 2s + t/3   [This will cancel the leading coefficient in x]
• x = 3 - 2s + t/3 
• s = 0, t = 0   [You will substitute this in the equation]
• x = 3 - 2(0) + 0/3
• x = 3/3
• x = 1

Happy birthday! 🎉

I know you would not know this post but I would like to say happy birthday to you! Thank you for making the remaining days of school an infinity and even though I always make fun of you, because that's the only way for you to notice me, you don't get angry at me or anything. You're such a strong lady and you are oh, so fine. I hope you will remember me in these coming years. Happy birthday Therese, i love you. 

Solutions

   What do you think of when you see, or read, the word "solution"? Yes, it may be either in chemical equations or the answer to the problem. But solutions are applied, too, to Linear Algebra. The question is, what is a solution?

   A system of linear equations is a finite set of linear equations, each with the same variables. A solution of system of linear equations is a vector that is simultaneously a solution of each equation in the system. The solution set of a system of linear equations is the set of all solutions of the system.We will refer to the process of finding the solution set of a system of linear equations as “solving the system.”
  As I have mentioned in my previous post, there are three roads that leads to an answer, and there are three types of solutions that leads to the answer.

Solve the following systems of linear equations:
(a)x - y = 1
    x + y = 3

(b)x - y = 2
    2x - 2y = 4

(c)x - y = 1
    x - y = 3

Yes? No? Okay.
Solutions:
(a) Adding the two equations together gives 2x = 4, so x = 2, from which we find that y = 1. A quick check confirms that [2, 1] is indeed a solution of both equations. That this is the only solution can be seen by observing that this solution corresponds to the point of intersection (2, 1) of the lines with equations x + y = 1 and x + y = 3, as shown in the figure:

(b) The second equation in this system is just twice the first, so the solutions are the solutions of the first equation alone—namely, the points on the line x - y = 2. These can be represented parametrically as [2  t, t].

(c) Two numbers x and y cannot simultaneously have a difference of 1 and 3.

Thus,

• (a) is a consistent solution. Consistent solutions has at least one (1) solution.
• (c) is an inconsistent solution. Inconsistent solutions has no intersections at all/ is parallel to each other, and we cannot find the value of one of the variables
• (b) is a type of consistent solution wherein you set a parameter, so it is an infinite solution. A parameter are any counting, whole, an integer, real, rational, or irrational numbers that is used to substitute a term to limit the infinite solutions

Systems of Linear Equations

“Where shall I begin, please your Majesty?” he asked.“Begin at the beginning,” the King said, gravely, “and go on till you come to the end: then stop.” 

—Lewis Carroll: Alice’s Adventures in Wonderland, 1865

Before learning everything and anything, we might want to need the base in every of linear equations.

   The word trivial is derived from the Latin root tri (“three”) and the Latin word via (“road”). Thus, speaking literally, a triviality is a place where three roads meet. The same system of equations can be viewed in three different, yet equally important, ways—these will be our three roads, all leading to the same solution. 

   Recall that the general equation of a line, ax + by = c, and that the general equation of a plane, ax  by  cz  d. Equations of this form are called linear equations.

Now, what is linear equation?

Are these examples of linear equations?

YES!

It is important to note that, although in these examples (and in most applications) the coefficients and constant terms are real numbers

Are these examples of linear equations?

NO!

Thus, linear equations do not contain products, reciprocals, or other functions of the variables; the variables occur only to the first power and are multiplied only by constants.

linear algebra is totes cool, like oh my gauss