Identity matrix

Matrix can be considered as an array where elements are stored in the form of rows and columns. Here elements in matrix can be considered as any type like algebraic expression, symbols or numbers. Matrix can be treated as a useful way for representing the linear transformation in a simple manner. Values in a matrix can be represented as shown below.

Suppose we have some values like 1, 2, 3, 4, 5, 6, 7, 8 and 9.

We have a matrix of 3 * 3 size then it can be represent above given values in form of matrix as:

 1  2  3

 4  5  6

 7  8  9

Now we will discuss about one of the types of matrix which is known as identity matrix. According to the definition of Identity matrix we can say that it is a kind of matrix where I A = A and A I = A. If we want to simplify the definition of identity matrix then we can say that it is square matrix which has a value 1 in diagonal position of matrix from top left to down right and all remaining position carry value zero into it. Sometimes this matrix is also known as unit matrix. How a identity matrix looks like is given below:

                             1  0  0

Identity Matrix =          0  1  0

                             0  0  1

Some properties of identity matrix are described below:

A ) When unit matrix includes the product of two square matrix then it can be said that it generates the inverse of one another.

B ) When unit matrix is multiply by itself then it generates the same output.

C ) This kind of unit matrix has a positive definite square root value.

In mathematics, the concept of Math Order of Operations defines a priority level of any operation that describes which operation should be performed first when any mathematical expression carry multiple operations.

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Mathematical Induction

Mathematical induction is a process which can be applied to set of general statements for sets of positive integers or their connected sequences.

In this we find either a given statement is true for entire natural numbers or not. As we discussed that in the case of natural numbers we will include only positive numbers. This should not be interpreted as a form of inductive reasoning. The easiest form of induction characterizes that either a statement that includes a natural number 'n' holds for all values of 'n' is true or not.

It can be understood with help of an example:

Example: Show that sum of first 'n' odd integers is n²?

It can be written as 1 + 3 + 5 + 7 + ... + (2n - 1) = n² for all positive integers.

Let’s have a proof of above expression: First find the value of A (n). It is equals to A (n): 1 + 3 + 5 + 7 + ... + (2n - 1) = n², then in basic step we have to show A (1) is true. So it can be written as: Trivial: 1 = 1².

In Inductive step we need to show that A (n) is true for all 'n' therefore A (n + 1) is also true for all 'n'. Let A (n) is true so 1 + 3 + 5 + 7 + ... + (2n - 1) = n²,

Also write it as A (n + 1): 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)² follows:

 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)²,

It can also be written as:

n² + (2n + 1) = (n + 1)², this is the proof of mathematical induction.

Precipitation Reaction is used in formation of solid in a solution or inside another solid.

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Slope

In the previous post we have discussed about Proof of bay s theorem and In today's session we are going to discuss about,Slope of a line can be defined as ratio of change in vertical axis to change in horizontal among two points on a line. If the slope of line is undefined or not defined then it is known as vertical line and if the slope of line is given as 0, then it is known as horizontal line. Formula to find the slope equation is given as:

                   y = mx + c,

Here value of ‘m’ stands for slope of line and y- intercept is given by ‘c’. It is also described by using the formula given below:

m = k₁ – k₂ / l₁ – l₂,

Here ‘m’ stands for slope of line and a₁, a₂ are the points defined on y- axis and b₁, b₂ are the points defined on x- axis. Above equation can also be written as:

     m = k₂– k₁ / l₂ – l₁,

In this gradient of a line is also defined which is denoted by the given formula:

         m = tan θ,

Let’s take an example. It will be clear with help of an example:

Example 1: - Calculate slope of the line segment that join the points (4, -7) and (-7, 4)?

Solution: - As we see above that it is a line segment that join the points (4, -7) and (-7, 4). As we know that formula to find the slope is given as:

m = k₂ – k₁ / l₂ – l₁, here value of k₁ = 4, l₁ = -7 and k₂ = -7, l₂ = 4

Now put given values in formula to find its value. On putting value in formula to get result.

m = -7 – 4 / 4 – (-7), on further solving we get:

m = - 11 / 11,

So here we get the value of 'm' is -1. In this way we can find out the value of 'm'.

Mann Whitney Test can be used to see either two independent samples of observations are drawn from same distributions.

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Proof of bay s theorem

In mathematics, we will study different theorem. Here we will understand the concept of bay s theorem. Bay’s theorem can have two distinct interpretations. It is an important concept of Bayesian statistics and has different properties in the field of science and engineering. Formula that is used to solve the probability, that name was given after the 18 th – century by the great scientist British mathematician Thomas bayes. Now we talk about the formula used in Bay’s theorem which is given below :
K (I / J) = K (I ∩ J) / K (J) = K (I) * K (J | I) / K (J)
Now we will understand the proof of bay’s theorem: Suppose we have X and Y_j be two sets. Then the conditional probability requires that:
P (X ∩ Y_j) = P (X) P (Y_j | P), here the symbol ‘∩’ represented as intersection (‘and’) and also said :
P (X ∩ Y_j) = P (Y_j ∩ X) = P (Y_j) P (X | Y_j), therefore it can be written as:
P (Y_j ∩ X) = P (Y_j) P (X | Y_j) / P (X), now assume Z = ∪_{i = 1}ⁿX_i, so X_i is an event in Z and X_i ∩ X_j = ⱷ for i ≠ j, then we can write it as:
X = X ∩ Z = X ∩ (∪_{i = 1}ⁿX_i) = ∪_{i = 1}ⁿ(X ∩ X_i),
P (X) = P (∪_{i = 1}ⁿ(X ∩ X_i)) = ∑_{i = 1}^N P (X ∩ X_i), it can also be written as:
P (X) = ∑_{i = 1}^N P (X_i) P (X | X_i),
P (X_i | X) = P (X_i) P (X | X_i) / ∑_{i = 1}^N P (X_i) P (X | X_i) this is the proof of bay’s theorem.
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definition of a function

In the previous post we have discussed about How to Find the Domain of a Function and In today's session we are going to discuss about definition of a function. In this blog we will discuss the definition of a function. Function is used to show relationship among set of inputs and set of outputs in which every value of input is related to exactly one value of output. In other words a relation defined from I to J such that a sub set of I * J is called as function from I to J. Let's us understand the concept of function and its types. There are different types of function which are given as:

One – one function

Many one function

Onto function

Now we will have small introduction about all its types.

One – one function (it is also called as injection function): - A function f: I → J is said to be one – one function if every input value of element 'I' has different image in 'J'. So it can be written as:

f : I → J is one – one if value of 'i' not equal to 'j'. (i ≠ j) → f (i) ≠ f (j) for all ij Ԑ I.

Many one function: - A function f :I → J is said to be many one function if two or more elements of set 'I' have same images in 'J'. In mathematical form it can be written as:

f: I → J is a many one function if there exist a, b Ԑ I such that a ≠ b but f (a) = f (b).

Onto function: - A function f: I → J is said to be onto function or it is also said to be 'surjection' if every value of element 'J' is image of some element of 'I' that is if f (I) = J, and range of 'f' is co – domain of function 'f' or in other words elements of 'J' has no pre – image in element 'I'. This is all about types of functions.

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How to Find the Domain of a Function

Hello friends, in mathematics we will study about the range and domain of function. Here we will discuss how to find the domain of a function. Function can be defined as a tool used to demonstrate the relationship between values. Now we will see how to find the domain of a function. Generally, functions are defined as f (p) where ‘p’ is the value you assign it. Like, f (p) = p / 2 ('f' of 'p' is divided by 2) is a function, because for every value of 'p' you get another value 'p / 2. If we put different values of function 'p'. First of all if we put value of 'p' as 2 then we get:

⇒f (2) = 1,
⇒f (4) = 2.
To find domain of function it is necessary to learn what the domain of a function is. When we select the entire x - coordinates values in the given function, then these x- coordinates values are known as domain of a function. In same way, the possible ‘y’ coordinate values are said to be the range of a function. Suppose we have some values (6, -5), (-9, 4), (11, -9), (-15, 1), then domain of function can be found as:

The domain of a given function = 6, -9, 11, -15.

Range is all ‘y’ coordinate values, (know more about Domain of a function, here)

Range = -5, 4, -9, 1. Now we will discuss how to find domain of a function in details. We need to follow some steps to find domain of a function:

Step 1: First of all we have to assume a function that contains ‘x’ and ‘y’ coordinates.

Step 2: As we know the domain of a function is all ‘x’ coordinates values.

So, we can say that in a function if values of ‘x’ and ‘y’ coordinates are known then we can easily find the domain and range of a function. This is all about the domain of a function.

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tangent line approximation

In the previous post we have discussed about Newtons Law of Cooling Differential equation and In today's session we are going to discuss about tangent line approximation. The tangent line approximation in mathematics may be defined as approximating the value of any function that is hard to calculate with the help of the value of another function that is nearest to it.

To have more precise understanding of the tangent line approximation, let us take a very simple example. Consider a square root function f (x) that is f (x) = √x.

Now if we take x equal to 9 then f (9) = √9 which comes out as 3 and is very easily calculated. However if we consider x = 8.9 then f (8.9) = √8.9 and this cannot be calculated easily. Thus for f (8.9) = √(9 – 0.1), the value can be approximated with the help of the value which we got from f (9) = √9 = 3 ( since 0.1 is very small ).

Thus we can say that if the value of f (b + h) is hard to find whereas f ( b ) can be very easily calculated where | h | is very small (here the mod is taken as we need positive value of h) then the value of f (b+h) can be approximated with the help of the value of f (b). (know more about tangent line approximation, here)

Since in the example given above the value of h is equal to -0.1 thus the value of the function f can be approximated at the point (b+h) with the help of the tangent line to the graph of function f at the point b when | h | is very small.

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