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

Linear Algebra: What is what and How is How?

LINEAR ALGREBRA

• Is a branch of mathematics as old as mathematics itself
• The solving of the simple linear equation ax + b = 0 may be considered as the original problem of this subject*
• The study of systems of linear equations acquired a new significance after the creation of analytic geometry (by Réné Descartes)**

*Although this problem presents absolutely no difficulty at all, the method which solves it (i.e. if a ‚ 0, then the unique solution is given by x = (-b)/a, no solution if a=0 and b ‚ 0 and infinitely many solutions if both a and b are zero), together with the properties of the corresponding linear function y=ax+b, are fundamental models for the ideas and methods of all of linear algebra. For example, the basic idea behind the solution of a system of linear equations in several unknowns is that of replacing such a system by a chain of these simple equations (by “eliminating” the variables).

**It was possible to reduce all the fundamental questions about the arrangements of lines and planes in space to the investigation of such algebraic systems.

A BRIEF HISTORY OF LINEAR ALGREBRA

"The introduction and development of the notion of a matrix and the subject of linear algebra followed the development of determinants, which arose from the study of coefficients of systems of linear equations.  Leibnitz, one of the two founders of calculus, used determinants in 1693 and Cramer presented his determinant-based formula for solving systems of linear equations (today known as Cramer’s Rule) in 1750.  In contrast, the first implicit use of matrices occurred in Lagrange’s work on bilinear forms in the late 1700s.  Lagrange desired to characterize the maxima and minima of multivariate functions. His method is now known as the method of Lagrange multipliers.   In order to do this he first required the first order partial derivatives to be 0 and additionally required that a condition on the matrix of second order partial derivatives hold; this condition is today called positive or negative definiteness, although Lagrange didn’t use matrices explicitly. 

            Gauss developed Gaussian elimination around 1800 and used it to solve least squares problems in celestial computations and later in computations to measure the earth and its surface (the branch of applied mathematics concerned with measuring or determining the shape of the earth or with locating exactly points on the earth’s surface is called geodesy.  Even though Gauss’ name is associated with this technique for successively eliminating variables from systems of linear equations Chinese manuscripts from several centuries earlier have been found that explain how to solve a system of three equations in three unknowns by ”Gaussian” elimination.  For years Gaussian elimination was considered part of the development of geodesy, not mathematics.  The first appearance of Gauss-Jordan elimination in print was in a handbook on geodesy written by Wilhelm Jordan.  Many people incorrectly assume that the famous mathematician Camille Jordan is the Jordan in ”Gauss-Jordan” elimination. 

            For matrix algebra to fruitfully develop one needed both proper notation and the proper definition of matrix multiplication.  Both needs were met at about the same time and in the same place.  In 1848 in England, J.J. Sylvester first introduced the term ”matrix,” which was the Latin word for womb, as a name for an array of numbers.  Matrix algebra was nurtured by the work of Arthur Cayley in 1855. Cayley studied compositions of linear transformations and was led to define matrix multiplication so that the matrix of coefficients for the composite transformation ST is the product of the matrix for S times the matrix for T.  He went on to study the algebra of these compositions including matrix inverses.  The famous Cayley-Hamilton theorem which asserts that a square matrix is a root of its characteristic polynomial was given by Cayley in his 1858 Memoir on the Theory of Matrices. The use of a single letter A to represent a matrix was crucial to the development of matrix algebra.  Early in the development the formula det(AB) = det(A)det(B) provided a connection between matrix algebra and determinants.  Cayley wrote ”There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants.”

            Mathematicians also attempted to develop of algebra of vectors but there was no natural definition of the product of two vectors that held in arbitrary dimensions.  The first vector algebra that involved a noncommutative vector product (that is, v x w  need not equal w x v) was proposed by Hermann Grassmann in his book Ausdehnungslehre (1844).  Grassmann’s text also introduced the product of a column matrix and a row matrix, which resulted in what is now called a simple or a rank-one matrix.  In the late 19th century the American mathematical physicist Willard Gibbs published his famous treatise on vector analysis.  In that treatise Gibbs represented general matrices, which he called dyadics, as sums of simple matrices, which Gibbs called dyads.  Later the physicist P. A. M. Dirac introduced the term ”bra-ket” for what we now call the scalar product of a ”bra” (row) vector times a ”ket” (column) vector  and the term ”ket-bra” for the product of a ket times a bra, resulting in what we now call a simple matrix, as above.  Our convention of identifying column matrices and vectors was introduced by physicists in the 20th century.

            Matrices continued to be closely associated with linear transformations. By 1900 they were just a finite-dimensional subcase of the emerging theory of linear transformations.  The modern definition of a vector space was introduced by Peano in 1888.  Abstract vector spaces whose elements were functions soon followed.

            There was renewed interest in matrices, particularly on the numerical analysis of matrices, after World War II with the development of modern digital computers.  John von Neumann and Herman Goldstine introduced condition numbers in analyzing round-off errors in 1947.  Alan Turing and von Neumann were the 20th century giants in the development of stored-program computers.  Turing introduced the LU decomposition of a matrix in 1948. The L  is a lower triangular matrix with 1’s on the diagonal and the U is an echelon matrix.  It is common to use LU decompositions in the solution of a sequence of systems of linear equations, each having the same coefficient matrix. The benefits of the QR decomposition was realized a decade later.  The Q is a matrix whose columns are orthonormal vectors and R  is a square upper triangular invertible matrix with positive entries on its diagonal.  The QR factorization is used in computer algorithms for various computations, such as solving equations and find eigenvalues." 

---by http://everydaylinear.tumblr.com/

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If you are reading this, I want to be your friend. William is the name, and eating is my game. I have read 42 books (excluding academic-inclined books) and I plan to read more. A stand-proud student of one of the most prestigious high schools in the whole island and a loser when it comes to math-related subjects. So GOD HELP ME RUN THIS BLOG *wailing whale aka me in the distance*

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