Linear A

Examples of Linear A in the following topics:

• Linear Approximation

  • 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 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.
  • Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix.

• Linear Equations

  • 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.
  • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
  • where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and $f$ is a given function of the same nature as y (called the source term).

• Using R2 to describe the strength of a fit

  • 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.
  • or about 25% in the data's variation by using information about family income for predicting aid using a linear model.
  • If a linear model has a very strong negative relationship with a correlation of -0.97, how much of the variation in the response is explained by the explanatory variable?

• Second-Order Linear Equations

  • 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:
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.

• The Equation of a Line

  • 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.
  • 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.
  • If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of $y$ and $X$ values.

• Zeroes of Linear Functions

  • 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 $x$.

• Effects of Linear Transformations

  • 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
  • will have a mean of bμ+A, a standard deviation of bσ, and a variance of b2σ2.
  • It should be noted that the term "linear transformation" is defined differently in the field of linear algebra.

• Nonhomogeneous Linear Equations

  • Nonhomogeneous second-order linear equation are of the the form: $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $f(t)$ is nonzero.
  • In the previous atom, we learned that a second-order linear differential equation has the form:
  • where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • However, there is a very important property of the linear differential equation, which can be useful in finding solutions.
  • Identify when a second-order linear differential equation can be solved analytically

• Linear Equations and Their Applications

  • Linear equations are those with one or more variables of the first order.
  • A linear equation is an algebraic equation that is of the first order—that is, an equation in which each term is either a constant or the product of a constant and a variable raised to the first power.
  • 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.
  • where a, b, c, and d are constants and x, y, and z are variables.
  • Consider, for example, a situation in which one has 45 feet of wood to use for making a bookcase.

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
  • 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.
  • Linear functions form the basis of linear algebra.
  • A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$.