LaTeX - A Crash Course

Jai Ganesh · 2024-08-09 17:59:12

Matrix - X

Row operations

There are three types of row operations:
* row addition, that is adding a row to another.
* row multiplication, that is multiplying all entries of a row by a non-zero constant;
* row switching, that is interchanging two rows of a matrix;

These operations are used in several ways, including solving linear equations and finding matrix inverses.

Submatrix

A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2.

The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.

A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain; this type of submatrix has also been called a leading principal submatrix.

Square matrix

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.

Main types

Name : Example with n = 3

Diagonal matrix

{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}

gives

.

Lower triangular matrix

{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}

gives

Upper triangular matrix

{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}

gives

.

Diagonal and triangular matrix

If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.

Identity matrix

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,

{\displaystyle \mathbf {I} _{1}={\begin{bmatrix}1\end{bmatrix}},\ \mathbf {I} _{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ \ldots ,\ \mathbf {I} _{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}}

gives

.

It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:

AI_n = I_mA = A

gives

for any m-by-n matrix A.

Jai Ganesh · 2024-08-11 18:12:57

Matrix - XI

Finding inverse matrix

{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)I}

gives

where adj(A) denotes the adjugate matrix, det(A) is the determinant, and I is the identity matrix. If det(A) is nonzero, then the inverse matrix of A is

{\displaystyle A^{-1}={\frac {1}{\det(A)}}\operatorname {adj} (A).}

gives

.

This gives a formula for the inverse of A, provided

det(A) \ \neq \ 0

gives

.

In fact, this formula works whenever F is a commutative ring, provided that det(A) is a unit. If det(A) is not a unit, then A is not invertible over the ring (it may be invertible over a larger ring in which some non-unit elements of F may be invertible).

Jai Ganesh · 2024-08-13 00:05:52

Cramer's rule

Geometric interpretation - I

Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns presented here.

Given the system of equations

{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}&=b_{2}\end{matrix}}}

gives

it can be considered as an equation between vectors

{\displaystyle x_{1}{\binom {a_{11}}{a_{21}}}+x_{2}{\binom {a_{12}}{a_{22}}}={\binom {b_{1}}{b_{2}}}.}

gives

.

The area of the parallelogram determined by

{\displaystyle {\binom {a_{11}}{a_{21}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}}

gives

is given by the determinant of the system of equations:

{\displaystyle {\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}.}

gives

.

Jai Ganesh · 2024-09-01 17:15:17

Cramer's rule

Geometric Interpretation - II

In general, when there are more variables and equations, the determinant of n vectors of length n will give the volume of the parallelepiped determined by those vectors in the n-th dimensional Euclidean space.

Therefore, the area of the parallelogram determined by

{\displaystyle x_{1}{\binom {a_{11}}{a_{21}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}}

gives

has to be

{\displaystyle x_{1}}

gives

times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by

{\displaystyle {\binom {b_{1}}{b_{2}}}=x_{1}{\binom {a_{11}}{a_{21}}}+x_{2}{\binom {a_{12}}{a_{22}}}}

gives

and

{\displaystyle {\binom {a_{12}}{a_{22}}}.}

gives

Equating the areas of this last and the second parallelogram gives the equation

{\displaystyle {\begin{vmatrix}b_{1}&a_{12}\\b_{2}&a_{22}\end{vmatrix}}={\begin{vmatrix}a_{11}x_{1}&a_{12}\\a_{21}x_{1}&a_{22}\end{vmatrix}}=x_{1}{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}}

gives

.
from which Cramer's rule follows.

Jai Ganesh · 2024-12-25 00:01:15

Arithmetico-geometric sequence

In mathematics, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression. The nth element of an arithmetico-geometric sequence is the product of the nth element of an arithmetic sequence and the nth element of a geometric sequence. An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of expected values in probability theory, especially in Bernoulli processes.

For instance, the sequence

\dfrac{0}{1}, \dfrac{1}{2}, \dfrac{2}{4}, \dfrac {3}{8}, \dfrac{4}{16}, \ \dfrac{5}{32}, \cdots

gives

is an arithmetico-geometric sequence. The arithmetic component appears in the numerator, and the geometric one in the denominator.

The label of arithmetico-geometric sequence may also be given to different objects combining characteristics of both arithmetic and geometric sequences. For instance, the French notion of arithmetico-geometric sequence refers to sequences that satisfy recurrence relations of the form

{\displaystyle u_{n+1}=ru_{n}+d}

gives

, which combine the defining recurrence relations

{\displaystyle u_{n+1}=u_{n}+d}

gives

for arithmetic sequences and

{\displaystyle u_{n+1}=ru_{n}}

gives

for geometric sequences. These sequences are therefore solutions to a special class of linear difference equation: inhomogeneous first order linear recurrences with constant coefficients.

The series summation of the infinite elements of this sequence has been called Gabriel's staircase and it has a value of 2.

The label of arithmetico-geometric sequence may also be given to different objects combining characteristics of both arithmetic and geometric sequences.

In general,

{\displaystyle \sum _{k=1}^{\infty }{k}{r^{k}}={\frac {r}{(1-r)^{2}}}\quad \mathrm {for\ } 0<r<1.}

gives

.

Jai Ganesh · 2024-12-25 14:15:06

Derivative - I

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.

Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

Definition:

As a limit

A function of a real variable f(x)} is differentiable at a point a of its domain, if its domain contains an open interval containing a, and the limit

{\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}

gives

exists.

This means that, for every positive real number

{\displaystyle \varepsilon }

gives

, there exists a positive real number

{\displaystyle \delta }

gives

such that, for every h such that

|h| < \delta

gives

and

{\displaystyle h\neq 0}

gives

then

{\displaystyle f(a+h)}

gives

is defined, and

{\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,}

gives

where the vertical bars denote the absolute value. This is an example of the

(\epsilon, \delta)

gives

-definition of limit.

Jai Ganesh · 2024-12-31 16:56:07

Derivative - II

Let f be the squaring function:

f(x) = x^{2}

gives

.
Then the quotient in the definition of the derivative is

{\displaystyle {\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.}

gives

The division in the last step is valid as long as

{\displaystyle h\neq 0}

gives

.
The closer h is to 0, the closer this expression becomes to the value 2a. The limit exists, and for every input a the limit is 2a.
So, the derivative of the squaring function is the doubling function:

{\displaystyle f'(x)=2x

gives

.

Jai Ganesh · 2025-01-02 16:24:53

Derivative - III

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function f, specifically the points

{\displaystyle (a,f(a))}

gives

and

{\displaystyle (a+h,f(a+h))}

gives

.
As h is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of f at a. In other words, the derivative is the slope of the tangent.

Jai Ganesh · 2025-01-02 16:50:49

Derivative - IV

Using infinitesimals

One way to think of the derivative

{\textstyle {\frac {df}{dx}}(a)}

gives

is as the ratio of an infinitesimal change in the output of the function f to an infinitesimal change in its input. In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.[9] The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form

{\displaystyle 1+1+\cdots +1}

gives

for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the d in the Leibniz notation. Thus, the derivative of

{\displaystyle f(x)}

gives

becomes

{\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}

gives

for an arbitrary infinitesimal dx, where

{\displaystyle \operatorname {st} }

gives

denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real.

Jai Ganesh · 2025-01-02 22:41:50

Derivative - V

Rules of computation

In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as differentiation.

Rules for basic functions

The following are the rules for the derivatives of the most common basic functions. Here, a is a real number, and e is the base of the natural logarithm, approximately 2.71828.

Derivatives of powers:

\dfrac{d}{dx}x^a = ax^{a - 1}

gives

Functions, exponential

\dfrac{d}{dx}e^x = e^x

gives

.

Natural Logarithm

\dfrac{d}{dx}a^x = a^x(ln \ a) \ for \ a > 0

gives

\dfrac{d}{dx}ln(x) = \dfrac{1}{x} \ for \ x > 0

gives

\dfrac{d}{dx} \ {log}_a{x} = \dfrac{1}{x{ \ ln (a)}} \ for \ x, a > 0

gives

.

Math Is Fun Forum

#301 2024-08-09 17:59:12

Re: LaTeX - A Crash Course

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Re: LaTeX - A Crash Course

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Re: LaTeX - A Crash Course

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Re: LaTeX - A Crash Course

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