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