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Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a ...
An example of a linear polynomial in the variables , and is + + +. Linearity of a mapping is closely related to proportionality . Examples in physics include the linear relationship of voltage and current in an electrical conductor ( Ohm's law ), and the relationship of mass and weight .
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.. More precisely, if , …, are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between , …, is a sequence (, …,) of elements of R such that
For example, given a linear map T : ... There is a strong relationship between linear algebra and geometry, ... Practical Linear Algebra: A Geometry Toolbox, AK ...
The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is ...
For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x. Antisymmetric for all x, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false). [11] Asymmetric
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
This behavior is what produces the linear relationship when logarithms are taken of both () and , and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation.