Form Four Mathematics Notes Topic 7 Matrices And Transformations

Form Four Mathematics Notes Topic 7 Matrices And Transformations

Click Link Below To Join Our Groups

TELEGRAM | WHATSAPP

Click Here To Download Our Notes

 

Operations on Matrices

The Concept of a Matrix

Definition:

A matrix is an array or an Orderly arrangement of objects in rows and columns.

Each object in the matrix is called an element (entity).

Consider the following table showing the number of students in each stream in each form.

Form I II III IV
Stream A 38 35 40 28
Stream B 36 40 34 39
Stream C 40 37 36 35

From the above table, if we enclose the numbers in brackets without changing their arrangement, then a matrix is farmed, this can be done by removing the headings and the bracket enclosing the numbers (elements) and given a name (normally a capital letter).

Nowthe above information can be presented in a matrix form as

https://sdimg.blob.core.windows.net/images/User/19551/Original/28_1444994221034.png

Any matrix has rows and columns but sometimes you may find a matrix with only row without Colum or only column without row.

In the matrix A above, the numbers 38, 36 an 40 form the first column and 38, 35, 40 and 28 form the first row.

Matrix A above has three (3) rows and four (4) columns.

In the matrix A, 34 is the element (entity) in the second row and third column while 28 lies in the first row and fourth column. The plural form of matrix is matrices.

Normallymatrices are named by capital letters and their elements by small letters which represent real numbers.

https://sdimg.blob.core.windows.net/images/User/19551/Original/30_1445114527850.png

Order of a matrix (size of matrix)

The order of a matrix or size of a matrix is given by the number of its rows and the number of its columns.

So if A has m rows and n columns, then the order of matrix is m x n.

It is important to note that the order of any matrix is given by stating the number of its rows first and then the number of its columns.

https://sdimg.blob.core.windows.net/images/User/19551/Original/31_1445159078899.png

Types of matrices:

The following are the common types of matrices:-

https://sdimg.blob.core.windows.net/images/User/19551/Original/32_1445159476013.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/33_1445159511306.png

Matrices of order up to 2 X 2

When adding or subtracting one matrix from another, the corresponding elements (entities) are /added or subtracted respectively.

This being the case, we can only perform addition and subtraction of matrices with the same orders.

Example 4

Given that

https://sdimg.blob.core.windows.net/images/User/19551/Original/34_1445160365343.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/35_1445160393025.png

Matrices of order up to 2 X 2

Example 2

Given that

https://sdimg.blob.core.windows.net/images/User/19551/Original/36_1445160579084.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/37_1445160688121.png

Example 3

Solve for x, y and z in the following matrix equation;

https://sdimg.blob.core.windows.net/images/User/19551/Original/38_1445161042598.png

Exercise 1

Determine the order of each of the following matrices;

https://sdimg.blob.core.windows.net/images/User/19551/Original/39_1445161224845.png

2. Given that

https://sdimg.blob.core.windows.net/images/User/19551/Original/40_1445161317504.png

3. Given that

https://sdimg.blob.core.windows.net/images/User/19551/Original/41_1445161487012.png

4. A house wife makes the following purchases during one week: Monday 2kg of meat and loaf of bread Wednesday, 1kg of meat and Saturday, 1kg of meat and one loaf of bread. The prices are 6000/= per kg of meat and 500/= per loaf of bread on each purchasing day

a. Write a 3×2 matrix of the quantities of items purchased over the three days.

b. Write a 2×1 column matrix of the unit prices of meat and bread.

5. Solve for x, y and z in the equation

https://sdimg.blob.core.windows.net/images/User/19551/Original/42_1445161663788.png

Additive identity matrix.

If M is any square matrix, that is a matrix with order mxm or nxn and Z is another matrix with the same order as m such that

M+ Z= Z+M = M then Z is the additive identity matrix.

https://sdimg.blob.core.windows.net/images/User/19551/Original/43_1445162059730.png

The additive inverse of a matrix.

If A and B are any matrices with the same order such that A+B = Z, then it means that either A is an additive inverse of B or B is an additive inverse of A that is B=-A or A= -B

https://sdimg.blob.core.windows.net/images/User/19551/Original/44_1445162167427.png

Example 10

Find the additive inverse of A,

https://sdimg.blob.core.windows.net/images/User/19551/Original/45_1445162295139.png

Example 11

Find the additive identity of B if B is a 3×3 matrix.

https://sdimg.blob.core.windows.net/images/User/19551/Original/46_1445162416518.png

A Matrix of Order 2 X 2 by a Scalar

A matrix can be multiplied by a constant number (scalar) or by another matrix.

Scalar multiplication of matrices:

Rule: If A is a matrix with elements say a, b, c and d, or

https://sdimg.blob.core.windows.net/images/User/19551/Original/48_1445162610993.png

Example 5

Given that

https://sdimg.blob.core.windows.net/images/User/19551/Original/49_1445162769670.png

Solution;

https://sdimg.blob.core.windows.net/images/User/19551/Original/50_1445162880509.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/51_1445162905960.png

Example 6

Given,

https://sdimg.blob.core.windows.net/images/User/19551/Original/52_1445163185607.png

Solution;

https://sdimg.blob.core.windows.net/images/User/19551/Original/53_1445163298072.png

Two Matrices of order up to 2 X 2

Multiplication of Matrix by another matrix:

https://sdimg.blob.core.windows.net/images/User/19551/Original/54_1445164409212.png

AB is the product of matrices A and B while BA is the product of matrix B and A.

https://sdimg.blob.core.windows.net/images/User/19551/Original/55_1445164474481.png

In AB, matrix A is called a pre-multiplier because it comes first while matrix B is called the post multiplier because it comes after matrix A.

Rules of finding the product of matrices;

1. The pre –multiplier matrix is divided row wise, that is it is divided according to its rows.

2. The post multiplier is divided according to its columns.

3. Multiplication is done by taking an element from the row and multiplied by an element from the column.

4. In rule (iii) above, the left most element of the row is multiplied by the top most element of the column and the right most element from the row is multiplied by the bottom most element of the column and their sums are taken:

https://sdimg.blob.core.windows.net/images/User/19551/Original/56_1445164640825.png

Therefore it can be concluded that matrix by matrix multiplication is only possible if the number of columns in the pre-multiplier is equal to the number of rows in the post multiplier.

Example 7

Given That;

https://sdimg.blob.core.windows.net/images/User/19551/Original/57_1445164818899.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/58_1445164842578.png

From the above example it can be noted that AB≠BA, therefore matrix by matrix multiplication does not obey commutative property except when the multiplication involves and identity matrix i.e. AI=IA=A

Example 8

Let,

https://sdimg.blob.core.windows.net/images/User/19551/Original/60_1445165048904.png

Example 9

Find C×D if

https://sdimg.blob.core.windows.net/images/User/19551/Original/61_1445165147340.png

Product of a matrix and an identity matrix:

If A is any square matrix and I is an identity matrix with the same order as A, then AI=IA=A

Example 12

Given;

https://sdimg.blob.core.windows.net/images/User/19551/Original/62_1445165362633.png

Exercise 1

1. Given that A= (3 4) and

https://sdimg.blob.core.windows.net/images/User/19551/Original/63_1445166292434.png

2. If,

https://sdimg.blob.core.windows.net/images/User/19551/Original/64_1445166395030.png

3.Using the matrices

https://sdimg.blob.core.windows.net/images/User/19551/Original/68_1445166926952.png

4.Find the values of x and y if

https://sdimg.blob.core.windows.net/images/User/19551/Original/67_1445166727204.png

Inverse of a Matrix

The Determinant of a 2 X 2 Matrix

Determinant of a matrix

https://sdimg.blob.core.windows.net/images/User/19551/Original/69_1445172874695.png

Now the determinant of matrix A is then defined as the difference of the product of elements in the leading diagonal and the product of the elements in the main diagonal.

https://sdimg.blob.core.windows.net/images/User/19551/Original/70_1445173016873.png

Example 4

Find

https://sdimg.blob.core.windows.net/images/User/19551/Original/71_1445173128107.png

Example 5

Considering

https://sdimg.blob.core.windows.net/images/User/19551/Original/72_1445173261825.png

Example 6

Find the value of x

https://sdimg.blob.core.windows.net/images/User/19551/Original/73_1445173411212.png

Singular and non singular matrices:

Definition:

Asingular matrix is a matrix whose determinant is zero, while non – singular matrix is the one with a non zero determinant.

https://sdimg.blob.core.windows.net/images/User/19551/Original/74_1445173681271.png

Example 10

Find the value of y

https://sdimg.blob.core.windows.net/images/User/19551/Original/75_1445173828626.png

The Inverse of a 2 X 2 Matrix

Inverse of matrices

Definition: If A is a square matrix and B is another matrix with the same order as A, then B is the inverse of A if AB=BA=I where I is the identity matrix.

Thus AB=BA=I means either A is the inverse of B or B is the inverse of A.

https://sdimg.blob.core.windows.net/images/User/19551/Original/76_1445174062016.png

Where B=A-1, that is B is the inverse of matrix A

https://sdimg.blob.core.windows.net/images/User/19551/Original/77_1445174367874.png

Since we need the unknown matrix B, we can solve for p and q by using equations (i) and (iii) and we solve for r and s using equations (ii) and (iv)

https://sdimg.blob.core.windows.net/images/User/19551/Original/78_1445174521611.png

To get p proceed as follows

https://sdimg.blob.core.windows.net/images/User/19551/Original/79_1445174619275.png

Also to get r and s, the same procedure must be followed:

https://sdimg.blob.core.windows.net/images/User/19551/Original/80_1445174742777.png

And

https://sdimg.blob.core.windows.net/images/User/19551/Original/81_1445175408124.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/82_1445175451990.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/83_1445175487330.png

Note that, if |A|= 0, Then

https://sdimg.blob.core.windows.net/images/User/19551/Original/84_1445175722695.png

Example 11

Given that,

https://sdimg.blob.core.windows.net/images/User/19551/Original/85_1445176071950.png

Solution:

https://sdimg.blob.core.windows.net/images/User/19551/Original/86_1445176344734.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/87_1445176376201.png

Example 12

Which of the following matrices have inverses?

https://sdimg.blob.core.windows.net/images/User/19551/Original/88_1445176494407.png

Exercise 1

1. Find the determinant of each of the following matrices.

https://sdimg.blob.core.windows.net/images/User/19551/Original/89_1445176926072.png

2. Which of the following matrices are singular matrices?

https://sdimg.blob.core.windows.net/images/User/19551/Original/90_1445177073852.png

3. Find inverse of each of the following matrices.

https://sdimg.blob.core.windows.net/images/User/19551/Original/91_1445177328137.png

2 X 2 Matrix to Solve Simultaneous Equations

Solving simultaneous equations by matrix method:

https://sdimg.blob.core.windows.net/images/User/19551/Original/92_1445177538162.png

Now by equating the corresponding elements, the following simultaneous equations are obtained.

https://sdimg.blob.core.windows.net/images/User/19551/Original/93_1445177834388.png

Then B= A-1×C

https://sdimg.blob.core.windows.net/images/User/19551/Original/94_1445178013016.png

Example 5

By matrix method solve the following simultaneous equations:

https://sdimg.blob.core.windows.net/images/User/19551/Original/95_1445178255575.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/96_1445178306548.png

Multiplying A-1 an each side of the equation, gives,

https://sdimg.blob.core.windows.net/images/User/19551/Original/97_1445178471389.png

Example 6

Solve

https://sdimg.blob.core.windows.net/images/User/19551/Original/98_1445179233579.png

Multiplying A-1 on each side of the equation gives,

https://sdimg.blob.core.windows.net/images/User/19551/Original/99_1445179403633.png

Example 7

By using matrix method solve the following simultaneous equations:

https://sdimg.blob.core.windows.net/images/User/19551/Original/100_1445181277420.png

Multiplying A-1 on each side of the equation gives,

https://sdimg.blob.core.windows.net/images/User/19551/Original/101_1445181418839.png

Cramer’s Rule

https://sdimg.blob.core.windows.net/images/User/19551/Original/102_1445181692958.png

So

https://sdimg.blob.core.windows.net/images/User/19551/Original/103_1445181864286.png

Example 10

Find

https://sdimg.blob.core.windows.net/images/User/19551/Original/104_1445203074069.png

Example 11

By using Cramer’s rule

https://sdimg.blob.core.windows.net/images/User/19551/Original/105_1445182330668.png

Example 12

By using Cramer’s rule,

https://sdimg.blob.core.windows.net/images/User/19551/Original/106_1445182564796.png

Exercise 1

1. Use the matrix method to solve the following systems of simultaneous equations.

https://sdimg.blob.core.windows.net/images/User/19551/Original/107_1445182981478.png

Use Cramer’s rule to solve the following simultaneous equation

https://sdimg.blob.core.windows.net/images/User/19551/Original/108_1445183126213.png

3. Whythe system of simultaneous equations

https://sdimg.blob.core.windows.net/images/User/19551/Original/109_1445183432240.png

Matrices and Transformations

Definition: A transformation in a plane is a mapping which moves an object from one position to another within the plane. Figures on the plane can also be shifted from one position by a transformation.

A new position after a transformation on is called the image.

Examples of transformations are (i) Reflection (ii) Rotation (iii) Enlargement (iv)Translation.

Any Point P(X, Y) into P¹(X¹,Y¹) by Pre-Multiplying (ᵡᵧ) with a Transformation Matrix T

– Suppose a point P(x,y) in the x-y plane moves to a point P¢ (x¢,y¢) by a transformation T,

https://sdimg.blob.core.windows.net/images/User/19551/Original/31_1445322164303.png

A transformation in which the size of the image is equal that of the object is called an ISOMETRIC MAPPING.

The Matrix to Reflect a Point P(X, Y ) in the X-Axis

Reflection;

When you look at yourself in a mirror you seem to see your body behind the mirror. Your body is in front of the mirror as your image is behind it.

https://sdimg.blob.core.windows.net/images/User/19551/Original/32_1445322332886.png

An object is reflected in the mirror to form an image which is;

a. The same size as the object

b. The same distance from the mirror as the object

So reflection is an example of ISOMETRIC MAPPING.

The mirror is the line of symmetry between the object and the image.

Example 7

Find the image of the point A (2,3) after reflection in the x – axes.

Solution;

Plot point A and its image A¢ such that AA¢ crosses the x – axis at B and also perpendicular to it.

For reflection AB should be the same as BA¢ i.e. AB = BA¢

https://sdimg.blob.core.windows.net/images/User/19551/Original/33_1445324522933.png

From the figure, the coordinates of A ¢ are A¢ (2,-3). So the image of A (2,3) under reflection in the x-axis is A¢ (2,-3)

Normally the letter M is used to denote reflection and thus Mx means reflection in the x – axis.

So Mx(2,3) =- (2,-3).

https://sdimg.blob.core.windows.net/images/User/19551/Original/34_1445325619120.png

Where Mx means reflection in the x – axis and My means reflection in the y-axis.

The Matrix to Reflect a Point P(X, Y) in the Y-Axis

Example 2

Find the image of B(3,4) under reflection in the y- axis.

Solution:

From My (x.y)= (-x,y)

My (3 ,4 ) =( -3,4)

Therefore the image of B(3,4) is B'(-3,4) .

Reflection in the line y = x.

The line y=x makes an angle 45with x and y axes. It is the line of symmetry for the angle YOX formed by two axis. By using isosceles triangle properties, reflection of the point (1,0) in the line y=x will be ( 0,1) while the reflection of (0,2) in the line y=x will be ( 2, 0) it can be noticed that the coordinates are exchanging positions. Hence the reflection of the point (x,y) in the line y=x is ( y,x).

https://sdimg.blob.core.windows.net/images/User/19551/Original/35_1445331656884.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/36_1445331696970.png

Where My =xmeans reflection in the line y=x.

Example 6

Find the image of the point A(1,2) after reflection in the line y = x . Draw a sketch.

https://sdimg.blob.core.windows.net/images/User/19551/Original/37_1445333383693.png

Reflection in the line y = -x

The reflection of the point B(x,y) in the line y = -x is B'(-y,-x).

https://sdimg.blob.core.windows.net/images/User/19551/Original/38_1445335099584.png

Example 9

Find the image of B (3,4) after reflection in the line y=-x followed by another reflection in the line y=0.Draw a sketch.

Solution;

Reflection of B in the line y=-x is B'(-4,-3). The line y=0 is the x – axis. So reflection (-4,-3) in the x-axis is (-4,3)

Therefore the image of B (3,4) is B¢(-4,3).

The image of a point P(x,y) when reflected in the line making an angleαwith positive x-axis and passing through the origin.

If the line passes through the origin and makes an angle a with x – axis in the positive direction, then its equation is y= xtanα where tanαis the slope of the line.

Consider the following diagram.

https://sdimg.blob.core.windows.net/images/User/19551/Original/39_1445337125707.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/40_1445337543081.png

But OPQ is a right angled triangle.

So x = OP Cosβ and y = OPSinβ .

Again OP¢R is a right angled triangle and the angle P¢QR = a -β + a- β+ β, this is due to the fact that reflection is an isometric mapping.

Now the angle P¢OR = 2 a-β, then

https://sdimg.blob.core.windows.net/images/User/19551/Original/41_1445337913446.png

It follows therefore that if M is a reflection in the line inclined at a, then

https://sdimg.blob.core.windows.net/images/User/19551/Original/42_1445338490840.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/43_1445338521691.png

Example 18

Find the image of the point A (1, 2) after a reflection in the line y = x.

https://sdimg.blob.core.windows.net/images/User/19551/Original/44_1445338683636.png

Example 19

Find the image of B (3,4) after reflection in the line y = -x followed by another reflection in the line y = 0.

https://sdimg.blob.core.windows.net/images/User/19551/Original/46_1445338842971.png

But the line y = 0 has 0 slope because it is the x – axis,

https://sdimg.blob.core.windows.net/images/User/19551/Original/47_1445339016414.png

Example 20

Find the equation of the line y = 2x + 5 after being reflected in the line y = x,

Solution:

The line y = x has a slope 1

So tan a = 1 which means a = 450

To find the image of the line y = 2x + 5, we choose at least two points on it and find their images, then we use the image points to find the equation of the image line.

Now y = 2x + 5

https://sdimg.blob.core.windows.net/images/User/19551/Original/48_1445340987749.png

The points (0,5) and (1,7) lie on the line

https://sdimg.blob.core.windows.net/images/User/19551/Original/49_1445341184026.png

So the image line is the line passing through (5,0) and (7,1) and it is obtained as follows;

https://sdimg.blob.core.windows.net/images/User/19551/Original/50_1445341393622.png

Exercise 1

Self Practice.

1. Find the image of the point D (4,2) under reflection in the x – axis

2. Point Q (-4,3) is reflected in the y – axis. Find its image coordinates.

3. Reflect the point (5,4) in the line y = x

4. Find the image of the point (1,2) after a reflection in the line y = x followed by another reflection in the line y = -x.

5. Find the equation of the line y = 3x -1 after being reflected in the line x + y = 0.

A Matrix Operator to Rotate any Point P( X, Y ) Through 90° 180°, 270° and 360° about the Origin

Rotation:

Definition; A rotation is a transformation which moves a point through a given angle about a fixed point.

https://sdimg.blob.core.windows.net/images/User/19551/Original/11_1445254873781.png

Rotation is an isometric mapping and it is usually denoted by R.

Therefore Rθ means rotation of an object through an angleθ.

In the xy plane, whenθismeasured in the clockwise direction it is negative and when it is measured in the anticlockwise direction it is positive.

https://sdimg.blob.core.windows.net/images/User/19551/Original/12_1445255543174.png

Example 7

Find the image of the point P(1,0) after a rotation through 900 about the origin in the anti clockwise direction.

https://sdimg.blob.core.windows.net/images/User/19551/Original/13_1445255745956.png

P is on the x – axis, so after rotation through 900 about the origin it will be on the y – axis. Since P is 1unit from O, P¢ is also 1 unit from O, the coordinates of P¢ (0,1) are P¢ (0,1). Therefore R 900(1,0) = (0,1).

Example 8

Find the image of the point B (4,2) after a rotation through 900 about the origin in the anticlockwise direction.

Solution;

Consider the following figure,

https://sdimg.blob.core.windows.net/images/User/19551/Original/14_1445256380166.png

Exercise 1

Find the matrix of rotation through

a. 900 about the origin

b. 450 about the origin

c. 270about the origin

Find the image of the point (1,2) under rotation through 1800 ant –clockwise about the origin.

Find the image of the point (-2,1) under rotation through 2700 clockwise about the origin

Find the image of (1,2) after rotation of -900.

Find the image of the line passing through points a (-2,3) and B(2,8) after rotation through 900 clockwise about the origin

General formula for rotation

Consider the following sketch,

https://sdimg.blob.core.windows.net/images/User/19551/Original/20_1445257940077.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/21_1445258002572.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/22_1445258040585.png

Example 12

Find the image of the point (1,2) under a rotation through 1800 anticlockwise

https://sdimg.blob.core.windows.net/images/User/19551/Original/23_1445258395482.png

Therefore the image of (1, 2) after rotation through 1800 anticlockwise is (-1,-2).

Example 13

Find the image of the point (5,2) under rotation of 900 followed by another rotation of 1800 anticlockwise.

Solution:

https://sdimg.blob.core.windows.net/images/User/19551/Original/25_1445262150689.png

Therefore the image of (5,2) under rotation of 90followed by another rotation of 180anticlockwise is (2,-5) .

Translation

Definition: A translation is a mapping of a point P (x, y) into P’ (x’, y’) by the Vector (a, b) such that (x’, y’) = (x, y) + (a, b), translation is denoted by the letter T. So T maps a point (x, y) into x’, y’)

Where (x’, y’) = (x, y) + (a, b)

https://sdimg.blob.core.windows.net/images/User/19551/Original/26_1445262654793.png

Consider the triangle OPQ whose vertices are (0,0), (3,1) and (3,0) respectively which is mapped into triangle O¢P¢Q¢ by moving it 2 units in the positive x direction and 3 units in the positive y direction

https://sdimg.blob.core.windows.net/images/User/19551/Original/27_1445262827594.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/28_1445262857266.png

Example 18

If T is a translation by the vector (4,3), find the image of (1, 2) under this translation.

https://sdimg.blob.core.windows.net/images/User/19551/Original/29_1445263056857.png

Example 19

A translation T maps the point (-3, 2) into (4, 3). Find where (a) T maps the origin (b) T maps the point (7, 4).

https://sdimg.blob.core.windows.net/images/User/19551/Original/30_1445263246699.png

Example 20

Find the translation vector which maps the point (6,-6) into (7,16).

Solution

Given that (x, y) = (6,-6) and (x¢, y¢) = (7,16), (a, b) =?

From T (x, y) = (x, y) + (a, b) = (x’, y’),

then (7,16) = (6,-6)+(a,b) which means a=7-6 = 1 and b=16+6 = 22. Therefore translation vector (a,b) = (1,22).

The Enlargement Matrix E in Enlarging Figures

Definition: Enlargement is the transformation which magnifies an object such that its image is proportionally increases on decreased in size by some factor k. The general matrix of enlargement

https://sdimg.blob.core.windows.net/images/User/19551/Original/1_1445238823041.png

Example 3

Find the image of the square with vertices O(0,0), A (1,0), B (1,1) and C (0,1) under the

https://sdimg.blob.core.windows.net/images/User/19551/Original/2_1445239108139.png

https://sdimg.blob.core.windows.net/images/User/19551/Original/3_1445239143134.png

Example 4

Find the image of (6, 9) under enlargement by the matrix

https://sdimg.blob.core.windows.net/images/User/19551/Original/4_1445239309438.png

Example 5

Draw the image of a unit circle with center O (0,0) under

https://sdimg.blob.core.windows.net/images/User/19551/Original/5_1445239442178.png

Now the images of these points are (0,3), (3,0), (0,-3), (-3,0) and other points respectively, where the centre remains (0,0) and the radius becomes 3 units.

https://sdimg.blob.core.windows.net/images/User/19551/Original/6_1445239581192.png

I n the figure above, the circle with radius 1 unit and its image with radius 3 units C1 and C2 respectively are shown.

Linear Transformation:

Definition:

For any transformation T, any two vectors U and V and any real number t, T is said to be a linear transformation if and only if

T(t U) = tT(U) and T (U+V) = T(U) + T(V)

Example 10

Show that the rotation by 900about O(0,0) is a linear trans formation

Solution

Let U=(U1,U2) and V =(V1 , V2) be any two vectors in the plane and t be any real number

To show that R900 is the linear transformation we must show that

R900 (tU)= t R900 (U) and

R900 (U + V) = R900 (U) + R900 (V)

https://sdimg.blob.core.windows.net/images/User/19551/Original/7_1445240884034.png

Therefore, since R900 (U) + R900 (V) = R900 (U+V) and R900 (tU)= t R900 (U), then R900 is a linear trans formation.

Example 11

Suppose that T is a linear transformation such that

T(U) = (1,-2), T(V) = (-3,-1) for any vectors U and V, find

(a) T(U+ V) (b) T(8U) (c) T(3U -2V)

Solution

(a)Since T is a linear Transformation then

T( U+ V) = T(U) + T(V)

https://sdimg.blob.core.windows.net/images/User/19551/Original/8_1445241925098.png

Exercise 1

1. If

https://sdimg.blob.core.windows.net/images/User/19551/Original/9_1445247026213.png

2. Is the matrix of reflection in a line inclined at angle a, U=(6,1) , V=(-1,4) and a13500, find (a) m(U+V) (b) m(2V)

If U =(2,-7) and V=(2,-3), find the matrix of linear transformation T such that T(2U)=(-4,14) and T(3V) = (6,9)

4. What is the image of (1,2) under the transformation

https://sdimg.blob.core.windows.net/images/User/19551/Original/10_1445254268305.png

5. Given that I is the identify transformation such that I(U) =U for any Vector U, prove that I is a linear transformation.

 

Leave a Comment