18.14: Movie Scripts 15-16 - Mathematics

G.15 Kernel, Range, Nullity, Rank

Invertibility Conditions

Here I am going to discuss some of the conditions on the invertibility of a matrix stated in Theorem 16.3.1. Condition 1 states that (X = M^{-1}V) uniquely, which is clearly equivalent to 4. Similarly, every square matrix (M) uniquely corresponds to a linear transformation (L colon mathbb{R}^{n} ightarrow mathbb{R}^{n}), so condition 3 is equivalent to condition 1.

Condition 6 implies 4 by the adjoint construct the inverse, but the converse is not so obvious. For the converse (4 implying 6), we refer back the proofs in Chapter 18 and 19. Note that if (det M = 0), there exists an eigenvalue of (M) equal to (0), which implies (M) is not invertible. Thus condition 8 is equivalent to conditions 4, 5, 9, and 10.

The map (M) is injective if it does not have a null space by definition, however eigenvectors with eigenvalue (0) form a basis for the null space. Hence conditions 8 and 14 are equivalent, and 14, 15, and 16 are equivalent by the Dimension Formula (also known as the Rank-Nullity Theorem).

Now conditions 11, 12, and 13 are all equivalent by the definition of a basis. Finally if a matrix (M) is not row-equivalent to the identity matrix, then (det M = 0), so conditions 2 and 8 are equivalent.

Hints for Review Problem 3

Lets work through this problem. Let (L colon V ightarrow W) be a linear transformation. Show that (ker L={0_{V}}) if and only if (L) is one-to-one:

  1. item First, suppose that (ker L={0_{V}}). Show that (L) is one-to-one. Remember what one-one means, it means whenever (L(x) = L(y)) we can be certain that (x=y). While this might seem like a weird thing to require this statement really means that each vector in the range gets mapped to a unique vector in the range. We know we have the one-one property, but we also don't want to forget some of the more basic properties of linear transformations namely that they are linear, which means (L(ax+by) = aL(x) + bL(y)) for scalars (a) and (b). What if we rephrase the one-one property to say whenever (L(x) -L(y) = 0) implies that (x-y = 0)? Can we connect that to the statement that (ker L={0_{V}})? Remember that if (L(v) = 0) then (v in ker L={0_{V}}).
  2. Now, suppose that (L) is one-to-one. Show that (ker L={0_{V}}). That is, show that (0_{V}) is in (ker L), and then show that there are no other vectors in (ker L). What would happen if we had a nonzero kernel? If we had some vector (v) with (L(v) = 0) and (v ot = 0), we could try to show that this would contradict the given that L is one-one. If we found (x) and (y) with (L(x) = L(y)), then we know (x=y). But if (L(v) = 0) then (L(x) + L(v) = L(y)). Does this cause a problem?

G.16 Least Squares and Singular Values

Least Squares: Hint for Review Problem 2

Lets work through this problem. Show that (ker L={0_{V}}) if and only if (L) is one-to-one:

  1. First, suppose that (ker L={0_{V}}). Does this cause a problem?

How to Write a Script

This article was co-authored by Melessa Sargent. Melessa Sargent is the President of Scriptwriters Network, a non-profit organization that brings in entertainment professionals to teach the art and business of script writing for TV, features and new media. The Network serves its members by providing educational programming, developing access and opportunity through alliances with industry professionals, and furthering the cause and quality of writing in the entertainment industry.

There are 19 references cited in this article, which can be found at the bottom of the page.

wikiHow marks an article as reader-approved once it receives enough positive feedback. This article received 31 testimonials and 81% of readers who voted found it helpful, earning it our reader-approved status.

This article has been viewed 3,260,312 times.

Writing a script is a great way to stretch your creativity by making a short film, movie, or TV show. Each script starts with a good premise and plot that takes your characters on a life-changing adventure. With a lot of hard work and correct formatting, you can write your own script in just a few months!

18.14: Movie Scripts 15-16 - Mathematics

18 Then Jesus told his disciples a parable to show them that they should always pray and not give up. (A) 2 He said: “In a certain town there was a judge who neither feared God nor cared what people thought. 3 And there was a widow in that town who kept coming to him with the plea, ‘Grant me justice (B) against my adversary.’

4 “For some time he refused. But finally he said to himself, ‘Even though I don’t fear God or care what people think, 5 yet because this widow keeps bothering me, I will see that she gets justice, so that she won’t eventually come and attack me!’” (C)

6 And the Lord (D) said, “Listen to what the unjust judge says. 7 And will not God bring about justice for his chosen ones, who cry out (E) to him day and night? Will he keep putting them off? 8 I tell you, he will see that they get justice, and quickly. However, when the Son of Man (F) comes, (G) will he find faith on the earth?”

The Parable of the Pharisee and the Tax Collector

9 To some who were confident of their own righteousness (H) and looked down on everyone else, (I) Jesus told this parable: 10 “Two men went up to the temple to pray, (J) one a Pharisee and the other a tax collector. 11 The Pharisee stood by himself (K) and prayed: ‘God, I thank you that I am not like other people—robbers, evildoers, adulterers—or even like this tax collector. 12 I fast (L) twice a week and give a tenth (M) of all I get.’

13 “But the tax collector stood at a distance. He would not even look up to heaven, but beat his breast (N) and said, ‘God, have mercy on me, a sinner.’ (O)

14 “I tell you that this man, rather than the other, went home justified before God. For all those who exalt themselves will be humbled, and those who humble themselves will be exalted.” (P)

The Little Children and Jesus (Q)

15 People were also bringing babies to Jesus for him to place his hands on them. When the disciples saw this, they rebuked them. 16 But Jesus called the children to him and said, “Let the little children come to me, and do not hinder them, for the kingdom of God belongs to such as these. 17 Truly I tell you, anyone who will not receive the kingdom of God like a little child (R) will never enter it.”

The Rich and the Kingdom of God (S)

18 A certain ruler asked him, “Good teacher, what must I do to inherit eternal life?” (T)

19 “Why do you call me good?” Jesus answered. “No one is good—except God alone. 20 You know the commandments: ‘You shall not commit adultery, you shall not murder, you shall not steal, you shall not give false testimony, honor your father and mother.’ [a] ” (U)

21 “All these I have kept since I was a boy,” he said.

22 When Jesus heard this, he said to him, “You still lack one thing. Sell everything you have and give to the poor, (V) and you will have treasure in heaven. (W) Then come, follow me.”

23 When he heard this, he became very sad, because he was very wealthy. 24 Jesus looked at him and said, “How hard it is for the rich to enter the kingdom of God! (X) 25 Indeed, it is easier for a camel to go through the eye of a needle than for someone who is rich to enter the kingdom of God.”

26 Those who heard this asked, “Who then can be saved?”

27 Jesus replied, “What is impossible with man is possible with God.” (Y)

28 Peter said to him, “We have left all we had to follow you!” (Z)

29 “Truly I tell you,” Jesus said to them, “no one who has left home or wife or brothers or sisters or parents or children for the sake of the kingdom of God 30 will fail to receive many times as much in this age, and in the age to come (AA) eternal life.” (AB)

Jesus Predicts His Death a Third Time (AC)

31 Jesus took the Twelve aside and told them, “We are going up to Jerusalem, (AD) and everything that is written by the prophets (AE) about the Son of Man (AF) will be fulfilled. 32 He will be delivered over to the Gentiles. (AG) They will mock him, insult him and spit on him 33 they will flog him (AH) and kill him. (AI) On the third day (AJ) he will rise again.” (AK)

34 The disciples did not understand any of this. Its meaning was hidden from them, and they did not know what he was talking about. (AL)

A Blind Beggar Receives His Sight (AM)

35 As Jesus approached Jericho, (AN) a blind man was sitting by the roadside begging. 36 When he heard the crowd going by, he asked what was happening. 37 They told him, “Jesus of Nazareth is passing by.” (AO)

38 He called out, “Jesus, Son of David, (AP) have mercy (AQ) on me!”

39 Those who led the way rebuked him and told him to be quiet, but he shouted all the more, “Son of David, have mercy on me!” (AR)

40 Jesus stopped and ordered the man to be brought to him. When he came near, Jesus asked him, 41 “What do you want me to do for you?”

“Lord, I want to see,” he replied.

42 Jesus said to him, “Receive your sight your faith has healed you.” (AS) 43 Immediately he received his sight and followed Jesus, praising God. When all the people saw it, they also praised God. (AT)

Five number summary calculator

For five number summary calculation, please enter numerical data separated with comma (or space, tab, semicolon, or newline). For example: 288.3 322.8 870.9 979.7 140.9 -369.2 -318.9 356.4 957.6 -736.5 255.1 -120.8 741.6

Five number summary

A five number summary consists of these five statistics:

the minimum,
Q1 (the first quartile, or the 25% mark),
the median,
Q3 (the third quartile, or the 75% mark),
the maximum,

The five-number summary gives you a rough idea about what your data set looks like. For example, you’ll have your lowest value (the minimum) and the highest value (the maximum) or more concentrated data.
The main reason you’ll want to find a five-number summary is to find more useful statistics, like the interquartile range IQR, sometimes called the middle fifty.

How to enter data as a frequency table?

Simple. First-type data elements (separated by spaces or commas, etc.), then type f: and further write frequency of each data item. Each element must have a defined frequency that count of numbers before and after symbol f: must be equal. For example:

How to enter a grouped data?

Grouped data are data formed by aggregating individual data into groups so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data.

This grouped data you can enter:
10-20 20-30 30-40
f: 5 10 15

How to enter data as a cumulative frequency table?

Similar as frequency table, but instead f: type cf: in second line. For example:

The cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. The last value will always be equal to the total for all observations since all frequencies will already have been added to the previous total.

5. Gain representation.

In the business, you’re as good as your manager and/or agent. Period. The Catch-22 of this is that many of they won’t represent you until you don’t actually need them (i.e., when you’ve already made it.)

Then again, it’s really tough to make it without them. So how do you get a representative with some juice in Hollywood to take an interest in you and your work?

The days of finding representation via query letter are pretty much over, but some writers still find success through this route.

You could try make targeted inquiries to agents and managers and we have created the ultimate guide to getting an agent or manager which you should definitely check out. Most agents and managers, however, are secured via…

For New versions

Older versions of python may not have pip installed and get-pip will throw errors. Please update your python (2.7.15 as of Aug 12, 2018).

All current versions have an option to install pip and add it to the path.

If python is not in PATH, it'll throw an error saying unrecognized cmd. To fix, simply add it to the path as mentioned below[1].

Python 2.7 must be having pip pre-installed.

Try installing your package by:

  1. Open cmd as admin. ( win+x then a )
  2. Go to scripts folder: C:Python27Scripts
  3. Type pip install "package name" .

[1] Also note: You must be in C:Python27Scripts in order to use pip command, Else add it to your path by typing: [Environment]::SetEnvironmentVariable("Path","$env:PathC:Python27C:Python27Scripts", "User")


"Variable" comes from a Latin word, variābilis, with "vari(us)"' meaning "various" and "-ābilis"' meaning "-able", meaning "capable of changing". [3]

In the 7th century, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta. One section of this book is called "Equations of Several Colours". [4]

At the end of the 16th century, François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns. [5]

In 1637, René Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". [6] Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in a 1887 Scientific American article. [7]

Starting in the 1660s, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the infinitesimal calculus, which essentially consists of studying how an infinitesimal variation of a variable quantity induces a corresponding variation of another quantity which is a function of the first variable. Almost a century later, Leonhard Euler fixed the terminology of infinitesimal calculus, and introduced the notation y = f(x) for a function f , its variable x and its value y . Until the end of the 19th century, the word variable referred almost exclusively to the arguments and the values of functions.

In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere differentiable continuous function. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formal definition. The older notion of limit was "when the variable x varies and tends toward a , then f(x) tends toward L ", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

( ∀ ϵ > 0 ) ( ∃ η > 0 ) ( ∀ x ) | x − a | < η ⇒ | L − f ( x ) | < ϵ ,

in which none of the five variables is considered as varying.

This static formulation led to the modern notion of variable, which is simply a symbol representing a mathematical object that either is unknown, or may be replaced by any element of a given set (e.g., the set of real numbers).

It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general cubic equation

is interpreted as having five variables: four, a, b, c, d , which are taken to be given numbers and the fifth variable, x, is understood to be an unknown number. To distinguish them, the variable x is called an unknown, and the other variables are called parameters or coefficients, or sometimes constants, although this last terminology is incorrect for an equation, and should be reserved for the function defined by the left-hand side of this equation.

In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", " x is the variable of the function f: xf(x) ", " f is a function of the variable x " (meaning that the argument of the function is referred to by the variable x ).

In the same context, variables that are independent of x define constant functions and are therefore called constant. For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivative to obtain the other antiderivatives. Because the strong relationship between polynomials and polynomial function, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

This use of "constant" as an abbreviation of "constant function" must be distinguished from the normal meaning of the word in mathematics. A constant, or mathematical constant is a well and unambiguously defined number or other mathematical object, as, for example, the numbers 0, 1, π and the identity element of a group.

Other specific names for variables are:

  • An unknown is a variable in an equation which has to be solved for.
  • An indeterminate is a symbol, commonly called variable, that appears in a polynomial or a formal power series. Formally speaking, an indeterminate is not a variable, but a constant in the polynomial ring or the ring of formal power series. However, because of the strong relationship between polynomials or power series and the functions that they define, many authors consider indeterminates as a special kind of variables.
  • A parameter is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in mechanics the mass and the size of a solid body are parameters for the study of its movement. In computer science, parameter has a different meaning and denotes an argument of a function.
  • Free variables and bound variables
  • A random variable is a kind of variable that is used in probability theory and its applications.

All these denominations of variables are of semantic nature, and the way of computing with them (syntax) is the same for all.

Dependent and independent variables Edit

In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y , whose possible values depend on the value of another variable, say x . In mathematical terms, the dependent variable y represents the value of a function of x . To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y . For example, the state of a physical system depends on measurable quantities such as the pressure, the temperature, the spatial position, . and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.

Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent. [8]

The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation f(x, y, z) , the three variables may be all independent and the notation represents a function of three variables. On the other hand, if y and z depend on x (are dependent variables) then the notation represents a function of the single independent variable x . [9]

Examples Edit

If one defines a function f from the real numbers to the real numbers by

then x is a variable standing for the argument of the function being defined, which can be any real number.

the variable i is a summation variable which designates in turn each of the integers 1, 2, . n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).

In the theory of polynomials, a polynomial of degree 2 is generally denoted as ax 2 + bx + c, where a, b and c are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its polynomial function this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

In mathematics, the variables are generally denoted by a single letter. However, this letter is frequently followed by a subscript, as in x2 , and this subscript may be a number, another variable ( xi ), a word or the abbreviation of a word ( xin and xout ), and even a mathematical expression. Under the influence of computer science, one may encounter in pure mathematics some variable names consisting in several letters and digits.

Following the 17th century French philosopher and mathematician, René Descartes, letters at the beginning of the alphabet, e.g. a, b, c are commonly used for known values and parameters, and letters at the end of the alphabet, e.g. x, y, z, and t are commonly used for unknowns and variables of functions. [10] In printed mathematics, the norm is to set variables and constants in an italic typeface. [11]

For example, a general quadratic function is conventionally written as:

where a, b and c are parameters (also called constants, because they are constant functions), while x is the variable of the function. A more explicit way to denote this function is

which makes the function-argument status of x clear, and thereby implicitly the constant status of a, b and c. Since c occurs in a term that is a constant function of x, it is called the constant term. [12] : 18

Specific branches and applications of mathematics usually have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters. For example, the three axes in 3D coordinate space are conventionally called x, y, and z. In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist. A convention often followed in probability and statistics is to use X, Y, Z for the names of random variables, keeping x, y, z for variables representing corresponding actual values.

There are many other notational usages. Usually, variables that play a similar role are represented by consecutive letters or by the same letter with different subscript. Below are some of the most common usages.

Scientific Maths - infinite precision Mathematics Library

  • Members
  • 1323 posts
  • Last active: Nov 14 2015 06:56 PM
  • Joined: 30 Jan 2013

Aim - Providing high precision Mathematical functions for infinetly large numbers which are not supported natively in AHK

Irrelative functions removed (Antilog, nthRoot, Roots..) and are now located at h-Functions.ahk

  • SM_Solve() - Solve expressions in variables and expressions in expressions (See documentation) . They can be infinetly large
  • Add, subtract ( via SM_Add () )infinetly large numbers . They can be + , - or decimals .
  • Multiply infinetly large numbers ( via SM_Multiply() ). They can be + , - or decimals .
  • Divide infinetly large numbers (via SM_Divide() ) . They can be + , - or decimals .
  • SM_UniquePmt() - Unique permutaion of a series with respect to a number
  • SM_fact() - factorial . uses Multiply() for high end numbers
  • SM_Mod() - Mod . Supports large dividends and Divisors
  • SM_Floor() - Floor . Supprots large numbers
  • SM_Round() - Round . Large Numbers
  • SM_Ceil() - Ceil
  • SM_Greater - Compare infinitely large numbers
  • SM_Prefect - Changes an infinetly large number to most suitable form
  • SM_toExp - Convert a number to Exponent form
  • SM_fromExp - Convert a sci number to real number
  • SM_Pow() - Power . Supprorts large powers and numbers
  • SM_e(n) - e to the power n

Please Note that by infinetly large numbers, it doesn't mean that the functions dont support small numbers . They support everything.

Please transfer numbers in functions that handle large numbers as strings like
Evaluate("23899999999999999999999999999999999999999999999999", "2389999999999999999999999999999238999")
Solve() supports Nesting via (. ) brackets , factorial via ! sign and power via ^ sign.

Frequently Used Miniwebtools:

If you like List of Prime Numbers, please consider adding a link to this tool by copy/paste the following code:

Do us a favor and answer 3 quick questions

Thank you for participating in our survey. Your input will help us to improve our services.


We spend much time and money each year so you can access, for FREE, hundreds of tools and calculators. This is made possible only thanks to the adverting on our site.

Please help us continue to provide you with free, quality online tools by turing off your ad blocker or subscribing to our 100% Ad-Free Premium version. For instructions on how to disable your ad blocker, click here.