Mathematics:
– Linear maps in mathematics satisfy additivity and homogeneity of degree 1 properties.
– Additivity implies homogeneity for rational α.
– Linearity extends to linear operators like the derivative.
– Linear polynomials of degree 1 are represented by straight lines.
– Linear algebra involves systems of linear equations.
Linear Polynomials:
– Represented by straight line graphs.
– Equations like y = mx + b have a slope and y-intercept.
– Linear polynomials over real numbers satisfy additivity and homogeneity with a constant term of 0.
– Functions with a non-zero constant term are called affine functions.
Boolean Functions:
– Linear Boolean functions have specific properties in Boolean algebra.
– Certain conditions in the truth table define a function as linear.
– Negation, logical biconditional, exclusive or, tautology, and contradiction are linear functions.
– Correspond to linear maps over Boolean vector spaces.
– Linearity is determined by the truth values assigned to arguments.
Physics:
– Linearity is a property of differential equations in physics, like Maxwell equations.
– Linear motion follows a straight line trajectory.
– Instruments exhibit linearity for consistent output with input changes.
– Human senses are highly nonlinear compared to instruments.
Electronics:
– Linear operating region of electronic devices ensures proportional output to input.
– Linear equipment like high fidelity audio amplifiers maintains waveform integrity.
– Linearity in electronics may be valid within a specific operating range.
– Linear filters and amplifiers provide accurate signal representations.
– Crucial for maintaining signal integrity in electronics.
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In mathematics, the term linear is used in two distinct senses for two different properties:
- linearity of a function (or mapping);
- linearity of a polynomial.
An example of a linear function is the function defined by that maps the real line to a line in the Euclidean plane R2 that passes through the origin. 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. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved.
Because a function such as is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.
The word linear comes from Latin linearis, "pertaining to or resembling a line".
