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Topic: Linear function

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	Linear function Summary
		The theory is called linear regression and allows the statistician to determine the slope and the y-intercept of the so-called "line of best fit." It also provides for the calculation of a number called the "correlation coefficient" that tells how close this best fit line actually comes to "capturing" the trend of the data.
		For instance, to approximate the value of a non-linear function at a given point on its curve, one can use the linear function which is tangent to the curve at a nearby known point.
		Linear functions always have as domain the set of all real numbers and a range of all real numbers.
	www.bookrags.com /Linear_function (1110 words)

	Linear programming - Wikipedia, the free encyclopedia
		In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear.
		Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region.
		In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in the worst case undecidable, and in many practical situations (those with bounded variables) NP-hard.
	en.wikipedia.org /wiki/Linear_programming (1319 words)

	Linear - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
		Namely, linearity of a differential equation means that if two functions f and g are solution of the equation, then their sum f+g is also a solution of the equation.
		In electronics, the linear operating region of a transistor is where the collector-emitter current is related to the base current by a simple scale factor, enabling the transistor to be use as an amplifier that preserves the fidelity of audio signals.
	www.vacilando.org /_cliextra/baghdadmuseumorg/includepage.php?title=Linear&action=edit (585 words)

	Linear function - Wikipedia, the free encyclopedia
		A linear function is a mathematical function of a certain form.
		The problem with this definition is that functions of the above form - despite their names - do not necessarily satisfy the conditions of a linear map.
		The constant m is often called the slope or gradient while c is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
	en.wikipedia.org /wiki/Linear_function (254 words)

	Algebra:Function graphing - Wikibooks, collection of open-content textbooks
		For a linear function, knowing any two different points on the line or knowing the slope and any one point on the line is enough to determine the line and identify it by an equation.
		Equations of the form y = c are linear functions of the general form y = m x + b where m = 0 and the constant c equals the y-intercept b (in the general form).
		The remaining expression, 2 x - 3, is a linear function of x, with a slope m = 2 and a y-intercept ordinate of -3.
	en.wikibooks.org /wiki/Algebra:Function_graphing (3771 words)

	Linear prediction - Wikipedia, the free encyclopedia
		Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples.
		In digital signal processing linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory.
		In system analysis (a subfield of mathematics), linear prediction can be viewed as a part of mathematical modelling or optimization.
	en.wikipedia.org /wiki/Linear_prediction (426 words)

	Earth Math
		If the slope of a linear function is 0, then the function is neither increasing nor decreasing, but is constant.
		This means that the graph of the linear function crosses the horizontal axis at the point (0, 250).
		A graph of the linear function is shown in Figure 2.
	earthmath.kennesaw.edu /main_site/review_topics/linear_functions.htm (515 words)

	Notes on Linear Progarmming - MAT 6570
		Linear programming consists in solving the linear problems mentioned above, and developing the algorithms and software able to find the optimal solution to a problem, if there is one, or to prove that no solution exists if that is the case.
		Each problem in Linear Optimization can differ in terms of the objective function (the objective can be either to minimize or maximize a function), as well as in terms of the linear constraints (equality constraints, inequality constraints, smaller or equal, strictly smaller...).
		Draw the (linear) objective function for some point in the feasible region and note that this corresponds to a "level curve" of all values in the feasible region which all have the same value for the objective function.
	www.mc.edu /campus/users/travis/syllabi/6570/linearprogramming.html (1290 words)

	Graphing and Linear Functions
		In this section, we will apply these principles to the identity function to obtain a deeper understanding of linear functions (i.e., functions which graph as straight lines).
		In the previous section, we were able to derive linear equations in a variety of situations.
		Specifically, if we were given a table or graph of a function where the values for inputs of 0 and 1 were clearly visible, it was an easy matter to write down the equations.
	campus.northpark.edu /math/PreCalculus/Functions/Graphing/Linear (2346 words)

	[No title]
		Investigation: Find two linear function f(x) and g(x) such that their product is tangent to each of f(x) and g(x) at two distinct points.
		A linear function will be defined as a function of the form y = mx + b, where m represents the slope of the line defined by the function and b represents the y-intercept.
		So for linear equations of the form y = mx +b, where m is either one or negative one it appears that for the two linear equations to intersect the parabola such that they are tangent to the parabola formed by their product, the lines must have a slope of one but have different directions.
	jwilson.coe.uga.edu /emt668/EMAT6680.2003.fall/Shiver/assignment1/Assignment1.html (860 words)

	Figure - Nature Neuroscience (Site not responding. Last check: 2007-10-21)
		In general, a function is linear if it can be written as a weighted sum of its input variables plus a constant.
		Linear functions form lines (one variable), planes (two variables, as shown here) or hyperplanes (more than two variables).
		The activity of the output unit is obtained by taking a linear sum of the basis function units.
	www.nature.com /neuro/journal/v3/n11s/fig_tab/nn1100_1192_F1.html (217 words)

	4.1.4.1. Linear Least Squares Regression
		The estimates of the unknown parameters obtained from linear least squares regression are the optimal estimates from a broad class of possible parameter estimates under the usual assumptions used for process modeling.
		Linear models with nonlinear terms in the predictor variables curve relatively slowly, so for inherently nonlinear processes it becomes increasingly difficult to find a linear model that fits the data well as the range of the data increases.
		This means that linear models may not be effective for extrapolating the results of a process for which data cannot be collected in the region of interest.
	www.itl.nist.gov /div898/handbook/pmd/section1/pmd141.htm (883 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The most commonly used regression model, % the ordinary linear regression, models y as a normal random variable, whose % mean is linear function of the predictors, b0 + b1*x1 +...
		x3,y3,z3,'r-', x3([k k]),y3([k k]),[0 z3(k)],'r:'); zlim([0 1]); xlabel('X'); ylabel('Y'); zlabel('Probability density'); grid on; view([-45 45]); %% % In a generalized linear model, the mean of the response is modeled as a % monotonic nonlinear transformation of a linear function of the % predictors, g(b0 + b1*x1 +...).
		Logistic regression is a special case of a generalized % linear model, and is more appropriate than a linear regression for these % data, for two reasons.
	www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/stats/glmdemo.m (1165 words)

	Linear functions - modeling (Site not responding. Last check: 2007-10-21)
		Function notation replaces y with f(x) (pronounced "f of x") and is written f(x) = 2x + 1.
		A linear function is a function that has the form f(x) = mx + b.
		A linear equation that would somewhat model this would be the equation C(t) = 10x + 20.
	www.geocities.com /CapeCanaveral/Launchpad/2426/page4.html (524 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-21)
		You can think of a function as a machine that takes in a number, does some sort of work on it, and puts it out as another number at the other end of the machine.
		In this case, you put in some number x; the function multiplies it by m, then adds b to the result, and puts out this new number, which is called f(x).
		A big difference between the function and the equation is that a function is only allowed to put out ONE number for each number that goes in.
	mathforum.org /library/drmath/view/60365.html (714 words)

	Math 2090 Chapter 1
		Examples of linear functions from R to R and from the plane to the plane.
		One should begin to grasp the notion that linearity for functions means that the operations of the function work in a predictable manner with the operations of addition and scalar multiplication; we see these stated as the condition for linearity, namely for a function f to be linear it must satisfy:
		Solutions to linear DE's can be expressed in the form xh(t) + xp(t), where xh(t) is called the homogeneous solution and satisfies the linear homogeneous DE, x'(t) + a(t)x(t) = 0 and xp(t) is called the particular solution and satisfies the original DE x'(t) + a(t) x(t) = f(t).
	www.math.lsu.edu /~forrest/m2090/chapter4.html (948 words)

	SchecterAdvancedMath
		Linear functions are ones where the output changes at a constant rate.
		If I give you any linear function there are dozens of ways to figure out the slope, and if you understand what slope means, then you can probably discover one yourself.
		Recall that the slope of a linear function is how much it goes up for every unit increase of input.
	www.stonehill.edu /compsci/schechter/SASSDS/lin.html (1160 words)

	Linear Functions (Site not responding. Last check: 2007-10-21)
		Linear functions are those whose graph is a straight line.
		A linear function has one independent variable and one dependent variable.
		Linear equations can be added together, multiplied or divided.
	www.columbia.edu /itc/sipa/math/linear.html (216 words)

	LINEAR TRANSFORMATIONS
		The central objective of linear algebra is the analysis of linear functions defined on a finite-dimensional vector space.
		is linear by properties of the trace of a matrix.
		In example 14 the kernel of the given linear transformation was a subspace of the domain.
	distance-ed.math.tamu.edu /Math640/chapter4/node2.html (880 words)

	Linear Functions
		Linear functions are the first type of functions one learns about in mathematics, yet there is not one single definition of linearity.
		Intuitively, linearity means proportionality of the output with respect to a variable.
		It is your responsibility that the function which you are drawing has at least part of its graph in the drawing area on the screen.
	web.mit.edu /org/a/aa-math/OldFiles/www/modules/node6.html (548 words)

	Analytical Placement: A Linear or a Quadratic Objective Function?
		The effect of a quadratic and a linear objective function on the chip area after final routing is demonstrated for benchmark circuits and other circuits with up to 21 000 cells.
		To minimize a linear objective function, linear programming methods [4] [5] and stochastic optimization techniques [6] have been used.
		The differences between linear and quadratic objective functions and their influence on the wiring are discussed in section 2.
	www.sigda.org /Archives/ProceedingArchives/Dac/Dac91/papers/1991/dac91/25_2/25_2.htm (2858 words)

	A linear function (Site not responding. Last check: 2007-10-21)
		where m and b are numbers represents a linear functions.
		When you write the function in this form, y = mx + b, then the slope is m.
		Thus the slope of this function is 9.
	mathcentral.uregina.ca /qq/database/QQ.09.02/chad1.html (205 words)

	Calculus I Notes, Section 2-9
		There are times when it is easier to find the derivative of a function that to find the value of a function.
		This fact, along with our knowledge of linear functions can prove to be a valuable tool in finding reliable approximations.
		Here is a view of the cube root function along with the linear approximation over the interval from (1,15).
	www.blc.edu /fac/rbuelow/calc/nt2-9.html (535 words)

	Pattern Classification Via Linear Programming
		A linear programming solution is attractive since many general purpose linear programming packages exist, thus requiring little or no implementation.
		Linear Programming: The problem of maximizing (or minimizing) a linear function subject to a finite number of linear constraints, also known as Operations Research, Optimization Theory, or Convex Optimization Theory.
		In 1964 Mosteller and Wallace used statistical inference and came to the conclusion that Madison was the author of all 12 disputed papers.
	cgm.cs.mcgill.ca /~beezer/cs644/main.html (1681 words)

	Linear Approximation
		The derivative supports a quick and effective means for approximating the values of complicated functions.
		A differentiable function is one for which there is a tangent line at each point on the graph.
		Of course, the graph of the linear function L
	oregonstate.edu /instruct/mth251/cq/Stage9/Lesson/linearApprox1.html (352 words)

	8.3 Linear Methods
		In this case the approximate value function is said to be linear in the parameters, or simply linear.
		For example, linear gradient-descent DP backups, using (8.3), with the on-policy distribution, will converge to the same result as TD(0).
		This is because the linear form prohibits the representation of interactions between features, such as the presence of feature i being good only in the absence of feature j.
	www.cs.ualberta.ca /~sutton/book/8/node4.html (715 words)

	Neural Networks
		In a linear model with sum squared error function, this error surface is a parabola (a quadratic), which means that it is a smooth bowl-shape with a single minimum.
		The simplest scaling function is minimax: this finds the minimum and maximum values of a variable in the training data, and performs a linear transformation (using a shift and a scale factor) to convert the values into the target range (typically [0.0,1.0]).
		Using the mean/SD scaling function with radial networks in regression problems, the training data is scaled so that its output mean corresponds to 0.0, with other values scaled according to the output standard deviation, and the bias is expected to be approximately zero.
	www.statsoft.com /textbook/stneunet.html (17723 words)

	Linear Approximation and Error Estimation
		We start with the observation that if you zoom in to a portion of a smooth curve near a specified point, it becomes indistinguishable from the tangent line at that point.
		The values of the function are close to the values of the linear function whose graph is the tangent line.
		For this reason, the linear function whose graph is the tangent line to y = f(x) at a specified point (a, f(a)) is called the linear approximation of f(x) near x = a.
	people.hofstra.edu /faculty/Stefan_Waner/RealWorld/calctopic1/linearapprox.html (835 words)

	Linear functions
		In other words, () serves as an example of a functional* equation - an equation whose unknown is a function.
		Continuity of the function is quite essential as it's possible to show [Ref. 1, 2] that the graph of any discontinuous solution to (*) is dense in the plane R
		A function is said to be linear if it's both additive and homogeneous.
	www.cut-the-knot.org /do_you_know/linear.shtml (329 words)

	On the convex hull of the composition of a separable and a linear function (Site not responding. Last check: 2007-10-21)
		On the convex hull of the composition of a separable and a linear function
		Theorems on the convex hull of an extended real-valued function living on a Hilbert space are presented in case the function is separable and in case the function is a composition of another function and a linear transformation.
		The results are applied to describe the convex hull of the objective function of a problem in stochastic integer programming.
	mally.eco.rug.nl /papers/chseplin.htm (97 words)

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