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Posts Tagged ‘linear’

Square Matrix

Tuesday, November 3rd, 2009

Introduction to Square Matrices

Description

A detailed tutorial on square matrices and how to identify them. Step by step tutorial including several examples of square matrices for reference.

Overview

A square matrix is a simple matrix in the shape of a square. It has the same number of rows and columns. Square matrices are called nxn matrces. The most common values for n are 2 and 3. Two columns and rows is the smallest amount of rows and columns a square matrix can have – matrices with only one value are not considered to be square.

Tags: 2, 2x2, 3, 3x3, algebra, columns, equal, equivalent, linear, matrices, matrix, n!, number, nxn, rows, same, shape, square, three, two, values
Posted in Algebra | No Comments »

Trace

Tuesday, November 3rd, 2009

How to Find the Trace

Description

A detailed tutorial on find the trace of a matrix. Step by step tutorial including several examples of how to find the trace for reference.

Overview

The trace of a square matrix is defined to be the sum of the elements on the main diagonal of the matrix. This can be mathematically expressed as:

$\mathrm{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{i i} \,$

Remember, the trace is only defined for square matrices – not any other kind of matrix.

Tags: algebra, diagonal, eigenvalue, element, invariant, linear, main, matrices, matrix, Spur, square, sum, trace
Posted in Algebra | No Comments »

Determinant

Tuesday, November 3rd, 2009

How to Find the Determinant

Description

A detailed tutorial on how to find the determinant. Step by step tutorial including several examples of finding the determinant for reference.

Overview

The determinant is a number that is associated with a square matrix. In a mathematical sense, the determinant is a scale factor for measure when the matrix is regarded as a linear transformation. The determinant is denoted by two bars on either side of the matrix, which can be confused with the absolute value of the matrix. The determinant is found by subtracting the products of the diagonals of the matrix, at least in a 2×2 matrix.

Tags: absolute, algebra, determinant, diagonal, factor, linear, matrices, matrix, product, scale, square, subtract, transformation, value
Posted in Algebra | No Comments »

Lorentz Transformations

Friday, October 30th, 2009

How to Solve Lorentz Transformations

Description

A detailed tutorial on Lorentz transformations. Step by step tutorial including several examples of Lorentz transformations for reference.

Overview

A Lorentz transformation is a way of describing how two different measurements of space and time can be converted into one frame of reference. This is because it was discovered that people who are moving at different velocities will report different times of certain events, or even a different order of events. The speed or velocity at which they are moving will throw things off. So by using a Lorentz transformation, you can get two different accounts to match up. Typically, a Lorentz transformation is a linear transformation.

Tags: algebra, events, frame, linear, Lorentz, measurement, order, reference, space, speed, time, transformation, velocity
Posted in Algebra | No Comments »

Scalar Triple Product

Tuesday, October 27th, 2009

Definition of a Scalar Triple Product

Description

A detailed tutorial on scalar triple products. Step by step tutorial including several examples of scalar triple products for reference.

Overview

A scalar triple product is a way of applying other multiplication operators to three vectors. Quite often, the scalar triple product is denoted as (a, b, c). It can also be defined as (a b c) = a(b x c). The scalar triple product has three main properties. The first one is that the absolute value of the scalar triple product is the volume of the three dimensional figure that is formed by the three vectors. The second one is the scalar triple product is only zero if the three vectors are linearly independent. The three vectors must lie in the same plane for this to be true. The third one is that the scalar triple product is only positive if all three of the vectors are considered right-handed.

A simple way to write the scalar triple product is to line up the coordinates of the vectors in this form: $(\mathbf{a}\ \mathbf{b}\ \mathbf{c})=\left|\begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{pmatrix}\right|.$ This is the same as saying $(\mathbf{a}\ \mathbf{b}\ \mathbf{c}) = (\mathbf{c}\ \mathbf{a}\ \mathbf{b}) = (\mathbf{b}\ \mathbf{c}\ \mathbf{a})= -(\mathbf{a}\ \mathbf{c}\ \mathbf{b}) = -(\mathbf{b}\ \mathbf{a}\ \mathbf{c}) = -(\mathbf{c}\ \mathbf{b}\ \mathbf{a}).$

Tags: absolute, algebra, box, coordinates, figure, independent, linear, mixed, multiplication, operator, parallelpiped, positive, product, properties, right-handed, scalar, three-dimensional, triple, value, zero
Posted in Algebra | No Comments »

Nonlinear Asymptotes

Thursday, October 22nd, 2009

How to Find Nonlinear Asymptotes

Description

A detailed tutorial on finding nonlinear asymptotes. Step by step tutorial including several examples of how to find nonlinear asymptotes for reference.

Overview

An asymptote is used to describe the behavior of a curve as it heads away from the origin and towards infinity. Typically it is meant to describe two curves that are doing this, and these curves are said to be asymptotic. In most cases, the asymptote is linear – which means the curves have the same behavior. Whenever someone is talking about an asymptote, they are talking about a linear asymptote unless they specify a different type of asymptote. In rare cases, asymptotes are nonlinear. Both curves are still heading towards infinity, but they do not have the same behavior. This can be determined by the limit of either the subtraction or the division of these curves.

Tags: algebra, asymptote, asymptotic, behavior, curve, division, function, horizontal, infinity, limit, linear, nonlinear, oblique, origin, subtraction, vertical
Posted in Algebra | No Comments »

Linear Regression

Friday, October 9th, 2009

Introduction to Linear Regression

Description

A detailed tutorial on linear regression. Step by step tutorial including several example problems of linear regression for reference.

Overview

Regression is a type of analysis that is used for analyzing several variables when the focus is on a dependent variable and one or more independent variables. Linear regression is when the dependent variable is a linear combination of the parameters. It can be used for both straight lines and parabolas, and each has a different formula.

Straight Line: $y_i=\beta_0 +\beta_1 x_i +\varepsilon_i,\quad i=1,\dots,N.\!$

Parabola: $y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\varepsilon_i,\ i=1,\dots,N.\!$

Tags: algebra, analyzing, combination, dependent, focus, independent, line, linear, Math, parabola, parameters, regression, straight, variable
Posted in Algebra | No Comments »

Oblique Asymptotes

Tuesday, October 6th, 2009

How to Find Oblique Asymptotes

Description

A detailed tutorial on how to find oblique asymptotes. Step by step tutorial including several examples of how to find oblique asymptotes for reference.

Overview

There are several different types of asymptotes. In this tutorial, we will be discussing oblique asymptotes. In order to find the oblique asymptotes of a function, you must first determine if the asymptote slants. If the numerator of a rational function has exactly one degree greater than the denominator, then the function slants and therefore has an oblique asymptote. When you divide the numerator and the denominator, the term or polynomial you get is the oblique asymptote.

Tags: algebra, asymptote, asymptotes, closer, curves, degree, denominator, distance, farther, function, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, numerator, oblique, origin, polynomial, positive, slant, straight, vertical, zero
Posted in Algebra | No Comments »

Horizontal Asymptotes

Tuesday, September 29th, 2009

How to Find Horizontal Asymptotes

Description

A detailed tutorial on how to find horizontal asymptotes. Step by step tutorial including several examples of how to find horizontal asymptotes for reference.

Overview

There are several different types of asymptotes. In this tutorial, we will be discussing horizontal asymptotes. In order to find the horizontal asymptotes of a function, take the limit of the function to infinity. Every function has a horizontal asymptote if it has a limit to infinity. The limit is your horizontal asymptote.

Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, oblique, origin, postive, straight, vertical, zero
Posted in Algebra | No Comments »

Vertical Asymptotes

Tuesday, September 29th, 2009

How to Find Vertical Asymptotes

Description

A detailed tutorial on how to find vertical asymptotes. Step by step tutorial including several examples of how to find vertical asymptotes for reference.

Overview

There are several different types of asymptotes. In this tutorial, we will be discussing vertical asymptotes. In order to find the vertical asymptotes of a function, we must first determine if there is a vertical asymptote. There is only a vertical asymptote if the limit of the function is equal to positive or negative infinity. If that is true, then the limit will reveal the vertical asymptote.