Sage-Tips

What are S and T in linear algebra?

If S and T are both linear transformations, S(x) = some vector y. Therefore T(y) is some other vector, say z.

How do you determine if an equation is dimensionally consistent?

A dimensionally consistent equation naturally takes the same form in all possible systems of units, since the same conversion factors are applied to both sides of the equation when transforming from one system to another.

Which equation represents exponential decay?

In mathematics, exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula y=a(1-b)x wherein y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed.

How do you find the equation of a wave?

1. To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form y(x,t)=Asin(kx−ωt+ϕ).
2. The amplitude can be read straight from the equation and is equal to A.
3. The period of the wave can be derived from the angular frequency (T=2πω).

How do you find the wave number?

The wave number for an EM field is equal to 2 pi divided by the wavelength in meters. (In some references, it is defined as the reciprocal of the wavelength in meters; in still others, it is defined as the reciprocal of the wavelength in centimeters.)

What is R4 in linear algebra?

The space R4 is four-dimensional, and so is the space M of 2 by 2 matrices. Vectors in those spaces are determined by four numbers.

Which equations are dimensionally consistent?

Dimension of x is [L], dimension of v is [L]/[T] and dimension of t is [T]. The dimensions on both the sides are the same. Thus, the equation x = vt is dimensionally consistent.

What are some examples of linear equations and constant functions?

Some of them are: Linear Equation General Form Example Intercept form x/a + y/b = 1 x/2 + y/3 = 1 As a Function f (x) instead of y f (x) = x + C f (x) = x + 3 The Identity Function f (x) = x f (x) = 3x Constant Functions f (x) = C f (x) = 6

What are the solutions of linear equations in one variable?

Solution of Linear Equations in One Variable 1 Clear the fraction x – 5 = 3 (x – 1) 2 Simplify Both sides equations 2x – 5 = 3x – 3 x = 3x + 2 x – 3x = 2 3 Isolate x

What is the difference between linear growth and exponential growth?

It may be helpful to compare linear growth and exponential growth. Consider the two functions whose graphs are shown in Figure173. L L is a linear function with initial value 5 5 and slope 2; 2; E E is an exponential function with initial value 5 5 and growth factor 2. 2.

What is the general form of the exponential function?

The general form of the exponential function is f(x) = abx, where a is any nonzero number, b is a positive real number not equal to 1. If b > 1, the function grows at a rate proportional to its size. If 0 < b < 1, the function decays at a rate proportional to its size.