MELJUN CORTES MATRIX ALGEBRA Rm104tr-10

7/29/2019 MELJUN CORTES MATRIX ALGEBRA Rm104tr-10

1/26

Lesson 10 - 1

Year 1

CS113/0401/v1

Matrix definition

Rectangular array of numbers

Size of matrix is given by no of

rows and no of columns e.g. A = 2 x 3 matrix

e.g. B = 3 x 3 matrix

2 9 16

1 0 -1

8 1 0

-1 0 1

4 1 5

LESSON 10MATRIX ALGEBRA


2/26

Lesson 10 - 2

Year 1

CS113/0401/v1

Vectors A single row matrix is called a

row Vector

e.g. [ 5 9 1 2 ]

A single column matrix is called

column vector

e.g.16

1

0

-1

MATRIX ALGEBRA


3/26

Lesson 10 - 3

Year 1

CS113/0401/v1

1 11 2

0 1 1

2 0 1

1 0 2

1 11 2

0 1 1

2 0 1

1 0 2

1+2 11+ 0 2+1

0+1 1+0 1+2

3 11 31 1 3

MATRIX OPERATION Matrix Addition

Must be of same dimension

result is of same dimension

E.g. A =

B =

A + B = +

=

=


4/26

Lesson 10 - 4

Year 1

CS113/0401/v1

1 11 2

8 1 1

-2 0 1

1 9 -2

1 11 2

8 1 1

-2 0 1

1 9 -2

1+(-2) 11+ 0 2-1

8-1 1-9 1-(-2)

3 11 17 -8 3

MATRIX OPERATION

Matrix Subtraction Same rule as matrix addition

e.g.A =

B =

A - B = +

=

=


5/26

Lesson 10 - 5

Year 1

CS113/0401/v1

5 2

1 -1

5 2

1 -1

5x2 2x21x2 -1x2

10 4

2 -2

MATRIX OPERATION

Matrix Multiplication Scalar Multiplication

e.g.A =

2A = 2

=

=


6/26

Lesson 10 - 6

Year 1

CS113/0401/v1

MATRIX OPERATION

Matrix Multiplication No of columns is 1st matrix must

be equal no of rows in 2nd matrix

Result is of dimension No of rows in 1st matrix by no

of column in 2nd matrix

e.g. If A is of dimension 2 x 3

B is of dimension 3 x 1

Then R=A * B is defined

R is of dimension 2 x 1


7/26

Lesson 10 - 7

Year 1

CS113/0401/v1

R1 C1

3 1

2 47 4

A =

B =8 0 5 43 2 11 1

AB =3 12 47 4

8 0 5 43 2 11 1

3x8+1x3 3x0+1x2 3x5+1x11 3x4+1x12x8+4x3 2x0+4x2 2x5+4x11 2x4+4x17x8+4x3 7x0+4x2 7x5+4x11 7x4+4x1

=

=

27 2 26 13

28 8 54 1268 8 79 32

MATRIX OPERATION

Matrix Multiplication


8/26

Lesson 10 - 8

Year 1

CS113/0401/v1

In matrix algebra unity is any

square matrix whose top left to

bottom right diagonal consists of

1s where all the rest of the matrixconsists of zeros

Matrices are only equal where

they are the same size and havethe same elements in the same

place, i.e.

I =1 0

0 1or I =

1 0 0

0 1 0

0 0 1

or I =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0

0 1 00 0 1

1 0

0 1

UNITY MATRIX


9/26

Lesson 10 - 9

Year 1

CS113/0401/v1

UNITY MATRIX

As wit normal numbers where a number

multiplied by one equals itself (3 x 1 =

3) so with matrices, A matrix multiplied

by the unity matrix equals itself, i.e.

AI = A and IA = A

1 00 1

A =1 0

0 1for example

AI = x =1 62 3 1x1+6x0 0x6+0x30x1+1x2 0x6+1x3

=1 6

2 3

Similarly1 6

2 3IA =

1 0

0 1x 1x1+6x0 1x0+6x1

2x1+3x0 2x0+3x1=

= 1 6

2 3 thus proving that Al = IA = A

Note: The unit matrix, I, must always be square.


10/26

Lesson 10 - 10

Year 1

CS113/0401/v1

EQUIVALENT MATRIX

Two matrices are equal if and only if

their corresponding elements are

equal. For instance, if

then matrix A = matrix B

Example:

a. Find the values of x and y if A + B = C

b. Is BC + CB?

c. Evaluate 3B

A = 2 34 5

2 3

4 5and B =

Given A = x 21 y

B =, 3 -54 2

8 -3

5 0and C =


11/26

Lesson 10 - 11

Year 1

CS113/0401/v1

a. A +B = x 21 y

+ 3 -54 2

X+3 2-5

1+4 y+2=

= X+3 -35 y+2

X+3 -3

5 y+2= 8 -3

5 0

X-3 = 8 and y + 2 = 0

EQUIVALENT MATRIX

Solution:

Since A + B = C

Therefore x = 5, y = 2


12/26

Lesson 10 - 12

Year 1

CS113/0401/v1

b. BC =3 -5

4 2

8 -3

5 0

= 24-25 -9+0

32+10-12+0= -1 -9

42 -12

CB =3 -5

4 2

8 -3

5 0

=24-12 -40-6

15+0 -25+0

=12 -46

15 -25

c. 3B = 33 -5

4 2

= 3x3 3(-5)3x4 3x2

= 9-1512 6

EQUIVALENT MATRIX Solution:

Thus BC = CB


13/26

Lesson 10 - 13

Year 1

CS113/0401/v1

A group operates a chain of filling

stations in each of which are employed

cashiers, attendants and mechanics as

shown

The number of filling stations are

How many of the various types of staff are

employed in Southern England and inNorthern England?

Large stations

Medium stations

Small stations

Southern England

3

5

12

Northern England

7

8

4

Matrix B, i.e. 3 x 2

Types of filling station

Cashier

Attendants

Mechanics

Large

4

12

6

Medium

2

6

4

Small

1

3

2

Matrix A, i.e. 3 x 3

Exercise


14/26

Lesson 10 - 14

Year 1

CS113/0401/v1

Solution

A is a 3 x 3 matrix, B is a 3 x 2 matrix

therefore AB is feasible and will be a 3 x

2 matrix.

A x B = AB

4 2 1

12 6 3

6 4 2

3 7

5 8

12 4

X11 X12

X21 X22

X31 X32

X11 = (4x3) + (2x5) + (1x12) = 34

X12 = (4x7) + (2x8) + (1x4) = 48

X21 = (12x3) + (6x5) + (3x12) = 102

X22 = (12x7) + (4x5) + (3x4) = 144

X31 = (6x3) + (4x5) + (2x12) = 62X32 = (6x7) + (4x8) + (2x4) = 82

Cashiers

AttendantsMechanics

South

3

512

North

7

84

AB is


15/26

Lesson 10 - 15

Year 1

CS113/0401/v1

1

2

1 2 3 4 5

3

4

5

6

6

C

C

A B

A B

X

y

TRANSFORMATIONA transformation is an operation which

transform a point or a figure into another

point or figure.

TranslationA translation is a transformation which

moves all points in a place through the

same direction.

e.g. The triangle ABC has been transformed

onto the triangle ABC by a translation [ ]i.e. 3 squares to the right and 2 squares up in

the plane of the paper.


16/26

Lesson 10 - 16

Year 1

CS113/0401/v1

Point a is mapped onto A by a

translation , denoted by T.

Enlargement (E)

An enlargement with centre O, scale

factor k is a transformation which

enlarges a given figure by k times the

original size.

If k > O, the given figure and its image

are on the same side of the centre of

enlargement O.

If k > O, the given figure and its image

are on opposite sides of O.

3

2

X

y

1

1

+

+

T =X

y

3

2 =4

3

Translation


17/26

Lesson 10 - 17

Year 1

CS113/0401/v1

OA OB OC= = =

OA OB OC

o

C

C

A

A

BB

Under an enlargement,Area of image

= k2Area of Figure

Enlargement (E)The figure and its image after an

enlargement are similar, The scale factor K

If the image of a point (x,y) under a

transformation is the point itself i.e. (x,y),the point (x,y) is called an invariant point of

the transformation.

If a line is mapped onto itself under a

transformation, the line is said to be an

invariant linr under the transformation.


18/26

Lesson 10 - 18

Year 1

CS113/0401/v1

A reflection is a transformation which

reflects all points of a plane in a line( on the plane ) called the mirror line.

ABC is mapped onto ABC under a

reflection in the line XY which is the

perpendicular bisector of AA, BB OR CC.

Under a reflection, the figure and its image

are congruent.

Example:

x

AA

CC

BB

Y

Reflection


19/26

Lesson 10 - 19

Year 1

CS113/0401/v1

Rotation (R)

A rotation is a transformation which

rotates all points on plane about a fixed

point known as the centre of rotation,

6through a given angle in anti-clockwise

of clockwise direction.

The angle through which the points are

rotated is called the angle of rotation.

The triangle ABC is rotated about the

origin O through 90 in the anti-clockwise

direction, and mapped onto triangle ABC.


20/26

Lesson 10 - 20

Year 1

CS113/0401/v1

OBC is mapped onto OBC under a shear along

the x-axis with factor k.

OC 6

K = = = 2OC 3

Shearing (H)A shear parallel to the x-axis is a

transformation which moves a point (x,y)

parallel to the x-axis through a distance

ky, where k is the shear factor.


21/26

Lesson 10 - 21

Year 1

CS113/0401/v1

difference in y-coordinates of corresponding pointsk =

x-coordinates of original point

Shearing (H)

A shear parallel to the y-axis is a

transformation which moves a point (x,y)

parallel to the y-axis through a distance kx,

where k is the shear factor.

Stretching (S)

One way stretch

A stretch parallel to the x-axis is a

transformation which move a point

(x,y) parallel to the x-axis, through adistance ky, where k is the stretch

factor.

A stretch parallel to the y-axis is a

transformation which moves a point

(x,y) parallel to the y-axis through a

distance ky, where k is the stretchfactor.


22/26

Lesson 10 - 22

Year 1

CS113/0401/v1

distance of new point the invariant linek =

x-coordinates of original point

Shearing (S)

In the case of stretching parallel to the x-

axis, the invariant line is the x-axis.

In the case of stretching parallel to the y-

axis, the invariant line is the y-axis.


23/26

Lesson 10 - 23

Year 1

CS113/0401/v1

Shearing (S)

Two Way Stretch

If a figure is stretched parallel to the x-

axis as well as parallel to the y-axis, then

the stretch is called a two-way stretch.

Under a two-way stretch with h and k as

constants of stretch parallel to the x-axis

and y-axis respectively a point (x,y) is

mapped onto (hx,ky).


24/26

Lesson 10 - 24

Year 1

CS113/0401/v1

Example: Matrix represents a

transformation T.Given (x,y) is the image of the point (a,b)

under the transformation T, find x and y in

terms of a and b.

Solution: Write the ordered pairs, (a,b)

and (x,y) as column vectors:

Premultiply by the matrix ,

we get

Therefore, x = a + 3b, y = 2a - 5b

1 3

2 -5

a

band

x

y

a

b

1 3

3 -5

x

y =1 3

2 -5

ab

=1xa + 3xb

2xa + (-5)xb

a + 3b

2a -5b=

Shearing (S)


25/26

Lesson 10 - 25

Year 1

CS113/0401/v1

The matrix defines a

transformation T which maps the points

(a,b) onto ( a + 3b, 2a - 5b ).

Example:Find the coordinates of the

image of the point (-3,2) under

the transformation represented

by the matrix

Solution:Let the image of the point = (x,y)

Therefore the images of the point = (-11,-15)

1 3

2 -5

3 -15 0

X

y=

3 -1

5 0

-3

2

3x(-3) + (-1)x25x(-3) + 0x2=

=

=-9-2

-15+0

-11

-15

Stretching (S)


26/26

Year 1

CS113/0401/ 1

Example:Find the matrix of the

transformation which maps

(1,0 ) onto (4,1) and (0,1)

onto (3,2).

Solution:Let the matrix of transformation

=

(1,0) (4,1)

because (4,1) is the image of (1,0)

1

0

Stretching (S)

a b

c d

4

1

a bc d=

4

1=

a +0c +0

=ac

MELJUN CORTES MATRIX ALGEBRA Rm104tr-10

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