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Linear inequality


Linear inequalities in real numbers


When two expressions are connected by 'greater than' or 'less than' sign,we get an inequation.

When operating in terms of real numbers, linear inequalities are the ones written in the forms

f(x) < b \, or f(x) \leq b,

where f(x) is a linear functional in real numbers and b is a constant real number. Alternatively, these may be viewed as

g(x) < 0 \, or g(x) \leq 0,

where g(x) is an affine function.

The above are commonly written out as

a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < 0


a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq 0

Sometimes they may be written out in the forms

a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < b


a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq b

Here x_1,\ x_2,...,x_n are called the unknowns, a_{0},\ a_{1},\ a_{2},...,\ a_{n} are called the coefficients, and b is the constant term.




a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; \leq \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; \leq \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; \leq \;&&& b_m \\ \end{alignat} Here x_1,\ x_2,...,x_n are the unknowns, a_{11},\ a_{12},...,\ a_{mn} are the coefficients of the system, and b_1,\ b_2,...,b_m are the constant terms.

This can be concisely written as the matrix inequality:

Ax \leq b

where A is an m&times;n matrix, x is an n&times;1 column vector of variables, and b is an m&times;1 column vector of constants.


Not all systems of linear inequalities have solutions.

Interpretations and applications

The set of solutions of a real linear inequality constitutes a half-space of the n-dimensional real space, one of the two defined by the corresponding linear equation.

The set of solutions of a system of linear inequalities corresponds to the intersection of the half-planes defined by individual inequalities. It is a convex set, since the half-planes are convex sets, and the intersection of a set of convex sets is also convex. In the non-degenerate cases this convex set is a convex polyhedron (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or a polyhedral cone). It may also be empty or a convex polyhedron of lower dimension confined to an affine subspace of the n-dimensional space Rn.

Sets of linear inequalities (called constraints) are used in the definition of linear programming.

Linear inequalities in terms of other mathematical objects

When you graph a linear inequality, it will be on one side of a line. Also, when you mark points where the line crosses where the x and y axis cross each other you can make the rise over run, which will help you find slope. If slope is denoted by m and y-intercept by b, you can find m = \frac {y_2 - y_1}{x_2 - x_1} and b = \frac {x_2y_1 - x_1y_2} {x_2-x_1}, so long as x_1 \neq x_2. Such line is described by the equation y = mx + b.

The above definition requires well-defined operations of addition, multiplication and comparison, therefore the notion of a linear inequality may be extended to ordered rings, in, particular, to ordered fields.

Linear combination



A linear combination is the sum of some set of ordered pairs (here, vectors), each ordered pair weighed (multiplied) by some real number. For example, if you had the ordered pairs regarding age and height, and you had the set {(5,30), (10,40), (50,70)}, an instance of linear combination would be 4*(5,30)+6*(10,40)+3*(50,70) which would equal (230,570).

Suppose that K is a field (a set of real numbers) and V is a vector space over K. As usual, we call elements of Vvectors and call elements of K scalars. If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is

a_1 v_1 + a_2 v_2 + a_3 v_3 + \cdots + a_n v_n. \,

There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, like in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace"; however one could also say "two different linear combinations can have the same value" in which case the expression must have been meant. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient are not considered to give new linear combinations.



Examples and counterexamples


Let the field K be the set R of real numbers, and let the vector space V be the Euclidean spaceR3. Consider the vectors e1&nbsp;= (1,0,0), e2&nbsp;= (0,1,0) and e3&nbsp;= (0,0,1). Then any vector in R3 is a linear combination of e1, e2 and&nbsp;e3.


= a_1 (1,0,0) + a_2 (0,1,0) + a_3 (0,0,1) \,
= a_1 e_1 + a_2 e_2 + a_3 e_3. \,


Let K be the set C of all complex numbers, and let V be the set CC(R) of all continuousfunctions from the real lineR to the complex planeC. Consider the vectors (functions) f and g defined by f(t)&nbsp;:= eit and g(t)&nbsp;:= e&minus;it. (Here, e is the base of the natural logarithm, about 2.71828..., and i is the imaginary unit, a square root of &minus;1.) Some linear combinations of f and g&nbsp;are:

  • \cos t = \begin{matrix}\frac12\end{matrix} e^{i t} + \begin{matrix}\frac12\end{matrix} e^{-i t} \,
  • 2 \sin t = (-i ) e^{i t} + ( i ) e^{-i t}. \,

On the other hand, the constant function 3 is not a linear combination of f and g. To see this, suppose that 3 could be written as a linear combination of eit and e&minus;it. This means that there would exist complex scalars a and b such that aeit + be&minus;it = 3 for all real numbers t. Setting t = 0 and t = &pi; gives the equations a + b = 3 and a + b = &minus;3, and clearly this cannot happen. See From Yahoo Answers

Question:im in algebra 2 and i have a project. i have to draw a picture made of functions. i have to include all 6 parents functions. i know there is linear quadratic cubic root and square root. What are the other 2? please help!!! THANKS!!

Answers:probably the absolute value (which looks like a V) and the cube function (y = x^3)

Question:1.Linear Parent 2.Absolute value 3.Quadratic 4.Square root 5.Cubic 6.Rational/Reciprocal Tell At least 4 characteristics about each. For example: What is the general shape? How many x intercepts does it have? Where are these equations seen? What is the domain and range? Is the function even/odd/neither? Are there any interesting facts about this parent function?

Answers:You could have just added this in to the additional details.... Linear: f(x)=x this is a straight line (ie linear) it has on x intercept (0) these equations are found by graphic a constant speed on a time/distance graph. D is all real numbers and R is all real. Function is odd (ie symmetrical from (0,0) Absolute value: f(x)= |x| The general shape is a V, it as one x intercept at 0 (the parent function does but it can have up to two), the are seen when you are only looking for positive y answers (ie distance)..... D is any real numbers and R is anything greater than 0, the function is even (symmetrical from the y axis) Quadratic: f(x)=x^2 General shape is a U, parent only has one x intercept but can have 2, D is any real number and R is anything greater than 0.... Even Square Root: f(x)= sqrt(x) GS- Steep closer to the origin but it flattens out (relative to the x axis) as you get farther away. One x intercept at 0. Domain is anything greater than 0 and range is any positive real number. Neither odd nor even Cubic: f(x)=x^3 GS- Looks like an S that has been turned sideways and stretched (http://www.google.com/imgres?imgurl=http://library.thinkquest.org/2647/media/oddxxx.gif&imgrefurl=http://library.thinkquest.org/2647/algebra/ftevenodd.htm&usg=__BxjG9VWZU593Jp7jRgtxmGIqWoc=&h=317&w=365&sz=3&hl=en&start=0&sig2=c49qej0i_HjEhrwj6KVCVg&zoom=1&tbnid=fQob5GWIX1BdNM:&tbnh=126&tbnw=144&ei=xz6ZTPK4C4P88AaW89DvDw&prev=/images%3Fq%3Df(x)%253Dx%255E3%26um%3D1%26hl%3Den%26safe%3Doff%26biw%3D1440%26bih%3D673%26tbs%3Disch:1&um=1&itbs=1&iact=hc&vpx=128&vpy=74&dur=342&hovh=126&hovw=145&tx=90&ty=95&oei=xz6ZTPK4C4P88AaW89DvDw&esq=1&page=1&ndsp=31&ved=1t:429,r:0,s:0) One x intercept at 0,0 Domain is any real number and range is any real number... Function is odd Reciprocal: f(x)=1/x GS-http://www.mathsrevision.net/gcse/functions.php No x intercepts Domain is any real number besides 0 and range is any real number besides 0.... I hope THIS answers your question....

Question:Define a function for || f || = max{0
Answers:You must verify the three axioms of a norm. For any f and g in C[0,1] and any a in R, you should have: || a f || = |a| || f || || f + g || || f || + || g || || f || = 0 if and only if f = 0 These are all fairly easy. The first is a computation: || a f || = max( | a f(x) | ) = max( |a| |f(x)| ) = |a| max( |f(x)| ) = |a| || f || The second is an application of the normal triangle inequality: || f + g || = max( | f(x) + g(x) | ) max( | f(x) | + | g(x) | ) max( |f(x)| ) + max( |g(x)| ) = || f || + || g || The third is also fairly obvious -- if || f || = max( |f(x)| ) = 0, then |f(x)| 0 everywhere. That implies f=0. And from the definition, it is clear that ||0|| = 0. QED For part b, just notice that 3x+2 is an increasing function so || f || = f(1) = 5.


Answers:I don't believe so. They have different slopes, so there would have to be some sort of rotation or reflection.

From Youtube

Parent Functions :This lesson discusses some of the basic characteristics of linear, quadratic, square root, absolute value and reciprocal functions. This was a lesson created for the MCR3U Functions course in the province of Ontario, Canada.

Mathematics - Study of Functions 1: Gradient of a Linear Function :This is the 2nd lesson in the series, "Study of Functions 1." This lesson looks at all linear graphs that go through the origin. The parent function is the graph of y = x. It describes the need to change the parent graph to get any other linear graph. Source: Mindset Network