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A **linear** village or a chain village is a village that is also a **linear** settlement.

over a field F, a **linear** map (also called, in some contexts, **linear** transformation, **linear** mapping or **linear** operator) is a map

, the equation :ax+by+c=0 is a **linear** equation in the single variable y for every value of x. It has therefore a unique solution for y, which is given by :

A **linear** motor is functionally the same as a rotary electric motor with the rotor and stator circular magnetic field components laid out in a straight line. Where a rotary motor would spin around and re-use the same magnetic pole faces again, the magnetic field structures of a **linear** motor are physically repeated across the length of the actuator.

Systems of **linear** equations form a fundamental part of **linear** algebra. Historically, **linear** algebra and matrix theory has been developed for solving such systems. In the modern presentation of **linear** algebra through vector spaces and matrices, many problems may be interpreted in terms of **linear** systems.

For some sets of vectors v 1 ,...,v n , a single vector can be written in two different ways as a **linear** combination of them: :Equivalently, by subtracting these a non-trivial combination is zero: :

Let T be the **linear** transformation associated to the matrix M. A solution of the system is a vector : such that :T(X)=v,that is an element of the preimage of v by T.

When a bijective **linear** map exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), the two spaces are isomorphic. Because an isomorphism preserves **linear** structure, two isomorphic vector spaces are "essentially the same" from the **linear** algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in **linear** algebra is testing whether a **linear** map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.

For example, let :be a **linear** system.

Since the motor moves in a **linear** fashion, no lead screw is needed to convert rotary motion to linear. While high capacity is possible, the material and/or motor limitations on most designs are surpassed relatively quickly due to a reliance solely on magnetic attraction and repulsion forces. Most **linear** motors have a low load capacity compared to other types of **linear** actuators. **Linear** motors have an advantage in outdoor or dirty environments in that the two halves do not need to contact each other, and so the electromagnetic drive coils can be waterproofed and sealed against moisture and corrosion, allowing for a very long service life.

A **linear** programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are **linear** inequalities. The list of constraints is a system of **linear** inequalities.

This defines a function. The graph of this function is a line with slope -\frac ab and y-intercept -\frac cb. The functions whose graph is a line are generally called **linear** functions in the context of calculus. However, in **linear** algebra, a **linear** function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is **linear** only when

It follows from this matrix interpretation of **linear** systems that the same methods can be applied for solving **linear** systems and for many operations on matrices and **linear** transformations, which include the computation of the ranks, kernels, matrix inverses.

If that is possible, then v 1 ,...,v n are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of **linear** dependence or independence of an arbitrary set S of vectors.

**Linear** maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces

A finite set of **linear** equations in a finite set of variables, for example, or is called a system of **linear** equations or a **linear** system.

In 1945, E. Pugliese Carratelli first introduced the classification of **Linear** A and **Linear** B parallels. However, in 1961 W.C. Brice modified the Carratelli system that was based on a wider range of **Linear** A sources, but Brice did not suggest **Linear** B equivalents to the **Linear** A signs. Louis Godart and Jean-Pierre Olivier introduced in the 1985 Recueil des inscriptions en linéaire A (GORILA), based on E.L Bennett's standard numeration of the signs of **Linear** B, introduced a joint numeration of the **Linear** A and B signs. The Egyptian exercise in writing the names of Keftiu even informs us of another Minoan ethnic identity in the form of mÈd|d|m whose first element cannot be associated with the name Midas, since it was already labeled by a linearly inscribed Hagia Triada (or HT 41.4) dating to c. 1350 BCE. This Egyptian evidence of the Keftiu language essentially indicates that words in its vocabulary are Semitic, but in the language predominantly of the Luwians. One might conclude from this that a Semitic language was used in Minoan Crete as a lingua franca for a largely Luwian population.

Note that the knowledge to be represented in **linear** equations is very close to that in a proper **linear** belief functions, except that the former assumes a perfect correlation between X and Y while the latter does not. This observation is interesting; it characterizes the difference between partial ignorance and **linear** equations in one parameter — correlation.

Generalized **linear** model - **Linear** regression

For the normal distribution, the generalized **linear** model has a closed form expression for the maximum-likelihood estimates, which is convenient. Most other GLMs lack closed form estimates.

where may be nonlinear functions. In the above, the quantities ε i are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients, β j in a **linear** way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely :are **linear** functions of the β j .

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