Transformation: An Introduction

Transformation: An Introduction

Transformation in sense of mathematics is the change of position, shape, or size of a geometric figure. A figure before the transformation is called an object and after transformation is called the image of the original object.


Figure of Reflection, Rotation, Translation, Enlargement and Reduction


In transformation, every point A of an object maps with a unique and distinct point A’ of the image, so it is a one-to-one function. 


If T is a transformation and A’ is the image point of A under T, then we denote it by T(A) = A’ or by T: A A’ or simply by A A’. 


If another transformation brings the image A’ back to the object (pre-image) point A, then such a transformation is called the inverse of the transformation T and it is denoted by T-1.



Transformation sometimes leaves certain points unchanged. These points are said to be invariant under the given transformation. Any other features which are left unaltered by the transformation such as distance, angle, parallelism, etc. are also described as invariant.



Types of Transformation


If a transformation maps a point A to A itself, then it is called an Identity Transformation and it is denoted by I. So, I(A) = A. Identity transformation leaves all points unchanged.


If a transformation does not change the figure after transformation, then it called an Isometric Transformation. Isometric transformation maps a figure to a congruent image. It means object figure and image figure are congruent under the isometric transformation.


If a transformation changes the figure after transformation, then it is called a Non-Isometric Transformation. Non-isometric transformation maps a figure to a similar image. It means object figure and image figure are similar under non-isometric transformation.


There are four fundamental transformations:


A.   Reflection

B.   Rotation

C.   Translation

D.  Enlargement (or Reduction)


Reflection, Rotation, and Translation are isometric transformations and Enlargement (or Reduction) is a non-isometric transformation.


There are four fundamental transformations:   A. Reflection   B. Rotaion   C. Translation   D. Enlagement (or Reduction)   Reflection, Rotation and Translation are isometric transformations and Enlargement (or Reduction) is an non isometric transformation.


Study the following figures for the general concepts of Reflection, Rotation,  Translation, Enlargement, and Reduction. 



Reflection
Reflection

Rotation
Rotation

Enlargement
Enlargement

Reduction
Reduction


Visit the following pages for specific study on particular topics of transformation and related worked out examples:


- Reflection

- Rotation

- Translation

- Enlargement (or Reduction)



Do You Have Any Question?

 

If you have any queries regarding the topics, you can comment here below.

Vector Geometry

Vector Geometry

Vector Geometry: We can study different properties and relations relating to geometry with the help of vectors. Such a study is known as the Vector Geometry.

Here are 12 geometrical theorems and their proofs by vector method.

Product of Vectors

Product of Vectors

Product of Vectors: Product of vectors or multiplication of vectors can be performed by the following two ways:

  • Scalar Product or Dot Product
  • Vector Product or Cross Product

Vector Operations

Vector Operations

Vector Operations: The vectors can be added, subtracted or multiplied with one another. Addition of vectors, difference of vectors, multiplication of a vector by a scalar, product of vectors, etc. are called vector operations.

 

Types of Vectors

Types of Vectors

Types of Vectors: After we have learnt ‘Vectors and its Components’ and ‘Magnitude and Direction of a Vector’, we’ll learn the Types of Vectors here. Vectors are of different types according to their magnitude, direction, position with respect to other given vector, and by the method of representing vectors.

Following are the different types of vectors:


Magnitude and Direction of a Vector

Magnitude and Direction of a Vector

Magnitude of a Vector 

Magnitude of a vector is the length of its directed line segment from the initial point to the terminal point. So, the magnitude of a vector is a positive number which is the measure of the line segment representing that vector. Magnitude of a vector is also known as the modulus or the absolute value of the vector.

Vectors and its Components

Vectors and its Components

Vectors: Introduction

The quantities which can be measured are called physical quantities. Some physical quantities have magnitudes only but some physical quantities have magnitudes as well as directions.

Let us study the addition of some physical quantities. For example, if we have two rectangles of areas 6cm2 and 8cm2, the total areas of these two rectangles is (6 + 8)cm2 i.e. 14cm2.

Sum of areas of two rectangles of area 6cm2 and  8cm2 is equal to 14cm2.

If two forces 4N and 5N act on a body, what is the total force acting on the body, 9N or 1N? Its answer is uncertain unless their directions are known. If two forces have same direction the total force on the body is 4N + 5N = 9N.

Sum (or resultant force) of two forces 4N and 5N acting toward the same direction is 9N i.e. 4N + 5N = 9N.

If two forces have opposite direction the total force on the body is 5N – 4N = 1N.

Sum (or resultant force) of two forces 4N and 5N acting toward the opposite direction is 1N i.e. 5N + (-4N) = 1N.

In the calculation of some physical quantities, the direction has very important role.

Solving Equations by Matrix Method

Solving Equations by Matrix Method

Solving Equations by Matrix Method

The simultaneous equations in two variables can be solved by different methods: substitution method, cross multiplication method etc. Here, we deal with the method of solving linear equations of two variables using matrices.

Let us consider the following linear equations,

a1x + b1y = c1 ………………. (i)

a2x + b2y = c2 ………………. (ii)

Writing equations (i) and (ii) in the matrix form, we get

(■(a_1&b_1@a_2&b_2 ))(■(x@y)) = (■(c_1@c_2 )) i.e.  AX = B ………………….. (iii) Where  A = (■(a_1&b_1@a_2&b_2 )),  X = (■(x@y))  and  B = (■(c_1@c_2 )) Pre-multiplying both sides of equation (iii) by inverse of A i.e. A-1, we get A-1(AX) = A-1B Or,  (A-1A)X = A-1B Or,  IX = A-1B   [∵ A-1A = I] ∴  X = A-1B   [∵  IX = X]

Equating the corresponding elements of equal matrices X and A-1B, we get the solution.

Note: If the determinant |A| = 0, the system of simultaneous equations have no unique solution, therefore when |A| ≠ 0, the equations have unique solution.

 

Worked Out Examples

Example 1: Solve uning matrices: x – 2y = -7 and 3x + 7y = 5 Solution: Here, x – 2y = -7 …………. (i) 3x + 7y = 5 ………… (ii) Writing equations (i) and (ii) in matrix form, we get (■(1&-2@3&7))(■(x@y)) = (■(-7@5)) i.e. AX = B Where, A = (■(1&-2@3&7)), X = (■(x@y)) and B = (■(-7@5)) ∴ |A| = |■(1&-2@3&7)| = 7 + 6 = 13 ∴ A-1 = 1/(|A|) (■(d&-b@-c&a)) = 1/13 (■(7&2@-3&1)) = (■(7/13&2/13@(-3)/13&1/13)) Now, by using formula X = A-1B, we have X = A-1B i.e.  (■(x@y)) = (■(7/13&2/13@(-3)/13&1/13))(■(-7@5)) Or,  (■(x@y)) = (■(7/13×-7+2/13×5@(-3)/13×-7+1/13×5)) Or,  (■(x@y)) = (■(-3@2)) Comparing the corresponding elements, x = -3, y = 2


Example 2: Solve by matrix method  4/x + 3/y = 7 and  3/x + 2/y = 4. Solution: Here, 4/x + 3/y = 7 ………………….. (i) 3/x + 2/y = 4 …………………. (ii) Writing equations (i) and (ii) in matrix form, we get (■(4&3@3&2))(■(1/x@1/y)) = (■(7@4)) i.e.  AX = B Where, A = (■(4&3@3&2)), X = (■(1/x@1/y)) and B = (■(7@4)) ∴|A| = |■(4&3@3&2)| = 8 – 9 = -1 ∴ A-1 = 1/(|A|) (■(d&-b@-c&a)) = 1/(-1) (■(2&-3@-3&4)) = (■(-2&3@3&-4)) Now, by using formula X = A-1B, we have X = A-1B i.e. (■(1/x@1/y)) = (■(-2&3@3&-4))(■(7@4)) = (■(-14+12@21-16)) = (■(-2@5)) Comparing the corresponding elements, 1/x = -2  or, x = (-1)/2   1/y = 5  or, y = 1/5 ∴  The values of x and y are (-1)/2  and 1/5.

Example 3: If the cost of 17 kg sugar and 4 kg tea is Rs. 1110 and the cost of 8 kg sugar and 2 kg tea is Rs. 540. Find the cost of sugar per kg and tea per kg by matrix method.

Solution: Let,

Cost of sugar per kg = Rs. x

Cost of tea per kg = Rs. y

Then according to questions,

1st case, 17x + 4y = 1110 …………… (i)

2nd case, 8x + 2y = 540 ……………… (ii)

Writing the given equations in the matrix form, we get

(■(17&4@8&2))(■(x@y)) = (■(1110@540)) i.e. AX = B Where, A = (■(17&4@8&2)), X = (■(x@y)) and B = (■(1110@540)) ∴ |A| = |■(17&4@8&2)| = 34 – 32 = 2 ≠ 0 So, the given equations have unique solution. Now, A-1 = 1/(|A|) (■(d&-b@-c&a)) = 1/2 (■(2&-4@-8&17))  Now, X = A-1B i.e.  (■(x@y)) = 1/2 (■(2&-4@-8&17))(■(1110@540)) Or,  (■(x@y)) = 1/2 (■(2×1110-4×540@-8×1110+17×540)) Or,  (■(x@y)) = 1/2 (■(60@300)) = (■(30@150))

Comparing the corresponding elements, we get

x = 30, y = 150

Hence, The cost of sugar per kg = Rs. 30

           The cost of tea per kg = Rs. 150

 

You can comment your questions or problems regarding the solution of simultaneous equations by matrix method here.

Minors and Cofactors of a Matrix

Minors and Cofactors of a Matrix

Minors and Cofactors of a Matrix

Minors:

Let A be a square matrix and aij is the element in ith row and jth column of A. Then the minor of the element aij is the determinant of the matrix formed by omitting ith row and jth column of A. The minor of element aij is denoted by Mij.