Exponential function & Logarithmic function

Exponential function:

An exponential function is a mathematical function of the form f(x) = ab^x, where a and b are constants and b is the base of the exponential function.
Exponential functions exhibit exponential growth or decay, depending on whether b is greater than 1 or less than 1.
The exponential function is used in many areas of mathematics and science, including calculus, finance, physics, and biology.
The natural exponential function, denoted by e^x, is the exponential function with the base 'e'.
The exponential function is the inverse of the logarithmic function, and they can be used to solve equations involving logarithmic and exponential functions.

Logarithmic function:

A logarithmic function is a mathematical function of the form f(x) = logb(x), where b is the base of the logarithmic function and x is the argument.
Logarithmic functions are used to express the relationship between two quantities that are being multiplied or divided.
The natural logarithmic function, denoted by ln(x), is the logarithmic function with the base 'e'.
Logarithmic functions have several properties, including the product rule, quotient rule, power rule, and change of base rule.
Logarithmic functions are the inverse of exponential functions, and they can be used to solve equations involving exponential and logarithmic functions.

Important Formulas to Remember in these Exponential function & Logarithmic function

Exponential Function:

Definition: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Properties:
- $$e^{a+b} = e^ae^b$$
- $$\frac{d}{dx}e^x = e^x$$
- $$\int e^x\,dx = e^x + C$$

Logarithmic Function:

Definition: $$\ln x = \log_e x$$

Properties:
- $$\ln 1 = 0$$
- $$\ln e = 1$$
- $$\ln(xy) = \ln x + \ln y$$
- $$\ln\left(\frac{x}{y}\right) = \ln x - \ln y$$
- $$\ln(x^c) = c\ln x$$
- $$\ln e^x = x$$
- $$\frac{d}{dx}\ln x = \frac{1}{x}$$
- $$\int \frac{1}{x}\,dx = \ln |x| + C$$

Meaning of 'e':

'e' is a mathematical constant that is approximately equal to 2.71828.
The constant 'e' is used in many areas of mathematics, including calculus, number theory, and probability theory.
The value of 'e' is irrational and cannot be expressed as a finite decimal or a fraction.
'e' is the base of the natural logarithm function, denoted by ln(x), and is used extensively in solving problems involving exponential and logarithmic functions.
The constant 'e' is also used in the exponential growth and decay models, and in the formula for calculating compound interest.

Important Formulas to Remember in these Meaning of 'e':

Meaning of 'e': $$e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n$$

Definition of logarithm and its properties

Definition of logarithm

The logarithm of a positive number x to the base b is the exponent y to which b must be raised to obtain x.
The logarithm function is denoted by logb(x), where b is the base and x is the argument.

properties:

The logarithm function has several properties, including the product rule, quotient rule, power rule, and change of base rule.
The product rule states that the logarithm of the product of two numbers is equal to the sum of their logarithms.
The quotient rule states that the logarithm of the quotient of two numbers is equal to the difference of their logarithms.

Important Formulas to Remember in these logarithm

Logarithms:

Definition of Logarithm: $$\log_{a}(b) = c \iff a^c = b$$

Properties of Logarithm:

$$\log_{a}(1) = 0$$

$$\log_{a}(a) = 1$$

$$\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)$$

$$\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)$$

$$\log_{a}(b^c) = c\log_{a}(b)$$

$$\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$$

Logarithms

Logarithm definition and their formulas:

Logarithms are mathematical functions that are used to express the relationship between two quantities that are being multiplied or divided.
The logarithm of a number is the exponent to which another fixed value called the base must be raised to produce that number.
Logarithms have a wide range of applications in mathematics, science, engineering, and finance.
The most commonly used logarithms are the base-10 logarithm (common logarithm) and the natural logarithm (base e).
Logarithmic functions are the inverse of exponential functions, and they can be used to solve equations involving exponential functions

Logarithms Formulas:

Definition of Logarithm: $$\log_{a}(b) = c \iff a^c = b$$

Properties of Logarithm:

$$\log_{a}(1) = 0$$

$$\log_{a}(a) = 1$$

$$\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)$$

$$\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)$$

$$\log_{a}(b^c) = c\log_{a}(b)$$

$$\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$$

Partial Fractions - Resolving a given rational function into partial fractions

Partial fractions is a technique used to decompose a rational function into simpler fractions.
The rational function is a fraction where the numerator and the denominator are both polynomials.
The partial fraction decomposition involves breaking the rational function into a sum of simpler fractions with denominators that are linear factors or irreducible quadratic factors.
The technique involves finding the unknown coefficients in the partial fraction expression by equating the numerator of the rational function to the sum of the numerators of the partial fractions.
Partial fraction decomposition is used in integration of rational functions, solving differential equations, and signal processing.

Resolving a given rational function into partial fractions:

To resolve a given rational function into partial fractions, first factorize the denominator of the rational function into linear or irreducible quadratic factors.
Express the partial fraction decomposition in the form of unknown coefficients over linear or irreducible quadratic factors.
Equate the numerator of the rational function to the sum of the numerators of the partial fractions, and solve for the unknown coefficients using algebraic methods.
If there are repeated factors in the denominator, then use the denominators raised to increasing powers as denominators in the partial fraction decomposition.
If the degree of the numerator is greater than or equal to the degree of the denominator, then perform long division to obtain a proper fraction, and then resolve the proper fraction into partial fractions.

Important Formulas to Remember in these Partial Fractions

Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions of the form: $$\\frac{A}{x-a} + \\frac{B}{(x-a)^2} + \\cdots + \\frac{L}{(x-a)^k} + \\frac{R(x)}{Q(x)}$$ where $a$ is a root of $Q(x)$ of multiplicity $k$ and $A, B, \\ldots, L$ are constants.

Resolving a Given Rational Function into Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions by following the steps given in the algorithm.

Gauss Jordan method

The Gauss-Jordan method is a technique used to solve systems of linear equations by manipulating an augmented matrix.
The augmented matrix is a matrix that contains the coefficients of the system of linear equations and the constants in the form of a matrix.
The method involves using row operations to transform the augmented matrix into a row echelon form or reduced row echelon form.
The row echelon form is a matrix where the leading coefficient of each row is to the right of the leading coefficient of the row above it.
The reduced row echelon form is a matrix where each leading coefficient is equal to 1, and all other entries in the column containing the leading coefficient are zero.
The reduced row echelon form is unique for each matrix, and it can be used to solve the system of linear equations.

Matrix inversion method

Matrix inversion is a method for finding the inverse of a matrix.
The inverse of a matrix can be used to solve systems of linear equations, find the coefficients of a polynomial, and perform other mathematical operations.
The inverse of a matrix can be found using the adjoint of the matrix and the determinant of the matrix.
A matrix is invertible if and only if its determinant is nonzero.
Matrix inversion can be computationally expensive for large matrices, and other methods such as LU decomposition or Gaussian elimination may be preferred.

Important Formulas to Remember in these Matrix inversion method

Let A be a square matrix of order n and B be the identity matrix of order n, then the inverse of A is given by $$A^{-1} = \frac{1}{|A|} adj\ A$$
The system has no solution if and only if the reduced row echelon form of the augmented matrix has a row of the form: $$\begin{bmatrix}0 & 0 & \\cdots & 0 & b\end{bmatrix}$$ where $b$ is a nonzero constant.
The system has infinitely many solutions if and only if the reduced row echelon form of the augmented matrix has a row of the form: $$\begin{bmatrix}0 & 0 & \\cdots & 0 & 0\end{bmatrix}$$

Solutions by Crammer‘s rule

Definition:

Cramer's rule is a method for solving a system of linear equations in 3 variables using determinants.

Solutions by Cramer's rule:

Cramer's rule is a method for solving a system of linear equations using determinants.
Cramer's rule involves calculating the determinants of matrices obtained by replacing the coefficient matrix with column vectors of constants.
The solution of the system of equations is obtained by dividing the determinant of each matrix by the determinant of the coefficient matrix.
Cramer's rule can be used to find the solution of a system of linear equations in any number of variables.
Cramer's rule is computationally expensive and is not suitable for solving systems of equations with a large number of variables.

Important Formulas to Remember in these topic Solutions by Crammer‘s rule:

Let A be the coefficient matrix and B be the constant matrix, then $$x = \frac{|X|}{|A|},\quad y = \frac{|Y|}{|A|},\quad z = \frac{|Z|}{|A|}$$ where X, Y, and Z are the matrices obtained by replacing the respective column of A with B and |A|, |X|, |Y|, and |Z| are the determinants of A, X, Y, and Z, respectively.

System of linear equations in 3 variables

A system of linear equations is a set of two or more equations that contain variables that are linearly related.
A system of linear equations in 3 variables involves three variables and can be represented by three linear equations.
A system of linear equations can be solved using various methods, such as substitution, elimination, or matrix inversion.
A system of linear equations in 3 variables can have either a unique solution, no solution, or infinitely many solutions.

Important Formulas to Remember in these topic System of linear equations in 3 variables:

$$\begin{aligned} a_{11} x + a_{12} y + a_{13} z &= b_1 \\ a_{21} x + a_{22} y + a_{23} z &= b_2 \\ a_{31} x + a_{32} y + a_{33} z &= b_3 \end{aligned}$$

Adjoint and multiplicative inverse of a square matrix

Main Points:

Adjoint and multiplicative inverse of a square matrix:
The adjoint of a matrix is a matrix that is obtained by taking the transpose of the matrix of cofactors of the original matrix.
The adjoint matrix is used to calculate the inverse of a matrix.
The multiplicative inverse of a matrix is a matrix that, when multiplied by the original matrix, gives the identity matrix.
A square matrix is invertible if and only if its determinant is nonzero.
The inverse of a matrix can be used to solve systems of linear equations, find the coefficients of a polynomial, and perform other mathematical operations.

Formulas to remember

Adjoint of A: $$(adj\ A)_{ij} = (-1)^{i+j} M_{ji}$$
Multiplicative Inverse of A: $$A^{-1} = \frac{1}{|A|} adj\ A$$

Singular and nonsingular matrices

A matrix is singular if it cannot be inverted, and nonsingular if it can be inverted.
A matrix is singular if its determinant is zero.
A matrix is nonsingular if its determinant is not zero.
A matrix that is singular cannot be used to find unique solutions to a system of linear equations.
Nonsingular matrices are important in mathematics and are used in a variety of applications, such as solving systems of equations and transforming data.

Formulas:

A matrix A is singular if and only if |A| = 0.
A matrix A is nonsingular (or invertible) if and only if |A| is nonzero.

Laplace‘s expansion

Laplace's expansion is a method for finding the determinant of a matrix by breaking it down into smaller submatrices.
The method involves selecting a row or column of the matrix, calculating the determinant of submatrices obtained by removing that row and column, and then multiplying each determinant by the corresponding element in the selected row or column.
Laplace's expansion formula can be used recursively to calculate determinants of larger matrices by breaking them down into smaller submatrices.
This method is not efficient for large matrices and other methods like Gaussian elimination or LU decomposition are preferred.
Laplace's expansion is a useful tool in mathematics for solving problems that involve matrices

Formula $$|A| = \sum_{i=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} a_{ij} C_{ij}$$

Properties of determinant

The determinant of a matrix is a scalar value that encodes information about the linear transformation represented by the matrix.
The determinant of a square matrix can be calculated using various methods, including cofactor expansion, row or column operations, and LU decomposition.
If a matrix has a row or column of zeros, then its determinant is zero.
The determinant of a diagonal matrix is the product of its diagonal elements.
The determinant of a product of matrices is equal to the product of their determinants, i.e., det(AB) = det(A) det(B). However, det(A+B) is not equal to det(A) + det(B) in general.
The determinant of the transpose of a matrix is equal to the determinant of the original matrix, i.e., det(A) = det(A^T).
A matrix is invertible if and only if its determinant is nonzero. In other words, a matrix is singular if and only if its determinant is zero.
The absolute value of the determinant of a matrix represents the factor by which the matrix scales the area or volume of a geometric object, depending on the dimensionality of the matrix.

Determinant of a square matrix

The determinant of a matrix is a scalar value that encodes information about the linear transformation represented by the matrix.
The determinant of a square matrix can be calculated using various methods, including cofactor expansion, row or column operations, and LU decomposition.
If a matrix has a row or column of zeros, then its determinant is zero.
The determinant of a diagonal matrix is the product of its diagonal elements.
The determinant of a product of matrices is equal to the product of their determinants, i.e., det(AB) = det(A) det(B). However, det(A+B) is not equal to det(A) + det(B) in general.
The determinant of the transpose of a matrix is equal to the determinant of the original matrix, i.e., det(A) = det(A^T).
A matrix is invertible if and only if its determinant is nonzero. In other words, a matrix is singular if and only if its determinant is zero.
The absolute value of the determinant of a matrix represents the factor by which the matrix scales the area or volume of a geometric object, depending on the dimensionality of the matrix.

Important Formulas to remember

Determinant of a Square Matrix: $$|A| = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}$$

Minor and Cofactor of an Element

The minor of an element in a matrix is the determinant of the submatrix obtained by deleting the row and column containing that element.
The cofactor of an element in a matrix is the minor of that element multiplied by (-1)^(i+j), where i and j are the row and column indices of the element.
Cofactors are used to find the inverse of a matrix and to solve systems of linear equations.
The determinant of a matrix can be calculated using the cofactor expansion method, which involves expanding along any row or column.
The determinant of a matrix is zero if and only if the matrix is singular, i.e., it has no inverse.

Important Formulas:

Minor of an Element: $$M_{ij} = \begin{vmatrix} a_{11} & \cdots & a_{1,j-1} & a_{1,j+1} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{i-1,1} & \cdots & a_{i-1,j-1} & a_{i-1,j+1} & \cdots & a_{i-1,n} \\ a_{i+1,1} & \cdots & a_{i+1,j-1} & a_{i+1,j+1} & \cdots & a_{i+1,n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & \cdots & a_{n,j-1} & a_{n,j+1} & \cdots & a_{nn} \end{vmatrix}$$
Cofactor of an Element: $$C_{ij} = (-1)^{i+j} M_{ij}$$

Symmetric and Skew-Symmetric Matrices:

A square matrix is symmetric if it is equal to its transpose, i.e., A = A^T.
The diagonal elements of a symmetric matrix are always real.
A square matrix is skew-symmetric if it is equal to the negative of its transpose, i.e., A = -A^T.
The diagonal elements of a skew-symmetric matrix are always zero.
The sum of a symmetric matrix and a skew-symmetric matrix is always a square matrix.

Formulas for Both

Symmetric Matrix: $$A = A^T$$
Skew-Symmetric Matrix: $$A = -A^T$$

Transpose of a Matrix

The transpose of a matrix is obtained by flipping the rows and columns of the original matrix.
The dimensions of the transpose matrix are opposite to that of the original matrix.
The transpose of a transpose matrix is equal to the original matrix.
Transposition is a linear operation, meaning that (A+B)^T = A^T + B^T and (kA)^T = k(A^T), where A and B are matrices and k is a scalar.
The transpose of a product of matrices is equal to the product of their transposes in reverse order, i.e., (AB)^T = B^T A^T.

Important Formula in this topic

Transpose of a Matrix: $$A^T = [a_{ji}]$$

Algebra of Matrices

Algebra of Matrices Important Points:

Matrices can be added or subtracted element-wise if they have the same dimensions.
Scalar multiplication is performed by multiplying each element of a matrix by a scalar value.
Matrix multiplication involves multiplying each row of the first matrix by each column of the second matrix.
The product of two matrices is only defined when the number of columns in the first matrix is equal to the number of rows in the second matrix.
Matrix multiplication is not commutative, meaning the order of multiplication affects the result.

List of Important formulas to remember

Algebra of Matrices:

Addition: $$A + B = [a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}]$$

Subtraction: $$A - B = [a_{ij}] - [b_{ij}] = [a_{ij} - b_{ij}]$$

Scalar Multiplication: $$kA = [ka_{ij}]$$

Matrix Multiplication: $$AB = [c_{ij}] \text{ where } c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

Types of Matrices

Types of Matrices:

Square matrix: A matrix with an equal number of rows and columns.
Rectangular matrix: A matrix with a different number of rows and columns.
Diagonal matrix: A square matrix with all elements outside the diagonal equal to zero.
Identity matrix: A diagonal matrix with all diagonal elements equal to one.
Triangular matrix: A square matrix where all elements below or above the diagonal are zero.

List of Formulas to remember:

Types of Matrices:

Row Matrix: $$[a_1, a_2, ..., a_n]$$

Column Matrix: $$\begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix}$$

Square Matrix: $$A = [a_{ij}]_{n \times n}$$

Diagonal Matrix: $$A = [a_{ij}] \text{ where } a_{ij} = 0 \text{ for } i \neq j$$

Identity Matrix: $$I_n = [a_{ij}] \text{ where } a_{ij} = 1 \text{ for } i=j \text{ and } a_{ij} = 0 \text{ for } i\neq j$$

Upper Triangular Matrix: $$A = [a_{ij}] \text{ where } a_{ij} = 0 \text{ for } i > j$$

Lower Triangular Matrix: $$A = [a_{ij}] \text{ where } a_{ij} = 0 \text{ for } i < j$$

Definition of Matrix

Definition of Matrix:

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
The dimensions of a matrix are given by the number of rows and columns it has, often written as "m x n".
Matrices are used in various areas of mathematics, science, and engineering to represent data, equations, and transformations.
A scalar is a special type of matrix with just one element, often used to represent constants or coefficients.
Matrices can be added, subtracted, multiplied, and transformed in various ways to perform operations on the data they represent.