Examples of linearity in the following topics:
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- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
- A common form of a linear equation in the two variables $x$ and $y$ is:
- The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
- Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
- Linear differential equations are of the form:
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- A linear approximation is an approximation of a general function using a linear function.
- In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
- Linear approximations are widely used to solve (or approximate solutions to) equations.
- Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
- If one were to take an infinitesimally small step size for $a$, the linear approximation would exactly match the function.
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- In the previous atom, we learned that a second-order linear differential equation has the form:
- When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
- However, there is a very important property of the linear differential equation, which can be useful in finding solutions.
- Phenomena such as heat transfer can be described using nonhomogeneous second-order linear differential equations.
- Identify when a second-order linear differential equation can be solved analytically
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- Linear and quadratic functions make lines and parabola, respectively, when graphed.
- In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
- Linear functions may be confused with affine functions.
- Although affine functions make lines when graphed, they do not satisfy the properties of linearity.
- Linear functions form the basis of linear algebra.
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- A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
- Linear differential equations are of the form $Ly = f$, where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
- The linear operator $L$ may be considered to be of the form:
- The linearity condition on $L$ rules out operations such as taking the square of the derivative of $y$, but permits, for example, taking the second derivative of $y$.
- When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
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- A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
- The graph of a linear function is a straight line.
- Linear functions can have none, one, or infinitely many zeros.
- To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.
- To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
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- Linear equations are those with one or more variables of the first order.
- There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
- Linear equations can therefore be expressed in general (standard) form as:
- For example,imagine these linear equations represent the trajectories of two vehicles.
- Imagine these linear equations represent the trajectories of two vehicles.
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- In statistics, linear regression can be used to fit a predictive model to an observed data set of $y$ and $x$ values.
- In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
- Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
- A common form of a linear equation in the two variables $x$ and $y$ is:
- The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
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- We evaluated the strength of the linear relationship between two variables earlier using the correlation, R.
- However, it is more common to explain the strength of a linear fit using R2, called R-squared.
- If provided with a linear model, we might like to describe how closely the data cluster around the linear fit.
- The R2 of a linear model describes the amount of variation in the response that is explained by the least squares line.
- 7.21: About R2 = (−0.97)2 = 0.94 or 94% of the variation is explained by the linear model.
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- This section covers the effects of linear transformations on measures of central tendency and variability.
- Let's start with an example we saw before in the section that defined linear transformation: temperatures of cities.
- To sum up, if a variable X has a mean of μ, a standard deviation of σ, and a variance of σ2, then a new variable Y created using the linear transformation
- It should be noted that the term "linear transformation" is defined differently in the field of linear algebra.