IT3201 Mathematics for Computing II mc2ict.cmb.ac.lk

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PREPARING FOR THE
IT3201 Mathematics for Computing II
mc2_at_ict.cmb.ac.lk
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(No Transcript)
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After successfully completing this module
students should be able to
Apply the mathematical (matrices, linear
transformations, vectors and differentiation
integration) and statistical (random variables,
discrete, and continuous probability
distributions) concepts easily in some of the
modules covered in the second and third year of
the degree course.
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Applications
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BOOKS RECOMMENDED FOR READING AND REFERENCE

• Business Mathematics by Qazi Zameeruddin,V.K
  Khanna and S.K Bhambri (vikas publishing house)
• Higher Algebra by H.S. Hall and S.K. Knight
  (A.I.T.B.S publishers and distributors) Delhi
  India
• An In-Depth Study of Mathematics by Dr. A.B.
  Mathur, (Pitambar publishing company)

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BOOKS RECOMMENDED

• New Comprehensive Mathematics for O Level, by
  Greer (Stanley Thornes publishers Ltd)
• Schaums Outline Probability and Statistics, by
  Murray R. Spiegel, J. Schiller, R. A.
  Srinivasan,  2nd edition, 2000, McGraw Hill
• Introduction to statistics Concepts and
  applications, Anderson D. R., Sweeney J. D.,
  Williams T.A., 3rd edition    

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BOOKS RECOMMENDED

• Vector Algebra by Shanti Narayan, S.Chand and
  Company Ltd, New Delhi
• Calculus with Analytical Geometry by Howard Anton
  (John Wiley and Sons)
• Further Pure Mathematics by Bostock, Chandler and
  Rourke (English Language Book Society / Stanley
  Thornes
• A Text Book of Matrices by Shanti Narayan, S
  Chand and Company Ltd, New  Delhi

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• Examination Procedure 
• Examination Paper will consist of two
  parts.
• Part 1 2 Hour paper consisting of Multiple
  Choice Questions  
• Part 21 Hour Paper consisting of Structured
  Questions

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Coordinate Transformations
Let u ax by v cx dy
(A)
For the moment, let us write (u,v)
(x,y)
a b c d
Suppose there is a further transformation given
by p ? u ?v
q ?u ?v
(B)
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Using the same notation,

• ?
• ? ?

(p,q) (u,v)
Symbolically,

• ?
• ? ?

a b c d
(x,y)
(1)
(p,q)
If we work in detail p ?(axby)?(cxdy)(?a?c)x
(?b?d)y q ?(axby)?(cxdy)(? a ?c)x(? b?
d)y
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• ?
• ? ?

a b c d
(x,y)
(1)
(p,q)
Using same notation (p,q)
aa ßc abßd ?a?c ?b ?d
(x,y) (2)
Comparing (1) and (2),we are tempted to write
aaßc abßd ?a?c ?b?d
a ß ? d
a b c d

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We are compelled to resort to a row by column
multiplication to obtain the right hand side from
the left hand side.i.e.
(a ß)
(row1,column1)
a c
aaßc
row1
(column1)
(a ß) ab ßd
b d
(row1,column2)
row1
column2
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(? ?) ?a ?c (row2,column1)
a c
(column1)
row2
(? ?) ?b ?d
(Row2,column2)
b d

(column1)
row2
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Thus the best notation for (A ) will be
u v
a b c d
x y

And that for (B) will be
p q

• ?
• ? ?

u v

(3)
Mathematically entities indicates by the above
square brackets are called MATRICES (Singular
MATRIX) In (3) the matrix on the left hand side
(L.H.S) has 2 rows and 1 column. It is called a
2x1 matrix.
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• ?
• ? ?

Similarly is called a 2x2 matrix.
It is called a square matrix of order 2.

• This is generalized to define an m x n matrix as
  an entity consisting of numbers forming m rows
  and n columns.
• Thus there will be mn numbers in all.

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• Written explicitly, it will look like

a11 a12 a13 .a1n a21 a22 a23
.a2n am1 am2 am3.amn
It is important to note that the element in the
ith row and jth column is aij
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• If A and B denotes two matrices and the product
  AB is to have a meaning, the row by column
  multiplication rule (rows of A by columns of B)
  will compel us to have an equal number of
  elements in each row of A and each column of B.

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• Therefore the number of columns of A and the
  number of rows of B should be equal.
• Thus if A is an m x n matrix and AB is to be
  defined,B must have n rows.
• Thus B has to be an n x p matrix where p is the
  number of columns of B.

Then AB has the number of rows of A and the
number of columns of B.
(A)mxn (B)nxp (AB)mxp
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It is evident that even when AB is defined, BA
may not be defined. BA will be defined only when
pm and in this case BA will be an nxn matrix.
Since AB is now an mxm matrix, AB and BA are not
equal unless mn in which case, both are square
matrices of the same order. An example shows that
even then AB and BA can be different.
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• 1
• 0 0

• 0
• 1 0

• 0
• 0 0

For example

• 0
• 1 0

• 1
• 0 0

• 2
• 1 1

and

• In addition to matrix multiplication , it is
  useful to define matrix addition. For this, they
  must be compatible by having the same number of
  rows and the same number of columns.Addition is
  performed by adding the corresponding elements.

For example if A is 2x3 only another 2x3 matrix B
can be added to A
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b11 b12 b13 b21 b22 b23
a11 a12 a13 a21 a22 a23
If A
and B
Then
a11b11 a12b12 a13b13 a21b21 a22b22
a23b23
AB
Thus it is evident that when A and B are
compatible for addition , AB BA
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Simple Application of a Matrix
A producer has 3 machines.The machines are
operated in two shifts. If (a ij) represents the
product which was produced in the i th shift by
the j th machine, the daily production can be
given by 2 x 3 matrix as follows.
Machine
1 2
3 A Shift 1 Shift 2

• 300 100
• 350 250 150

Product from 1st shift, 2nd machine 300 units
Product from 2nd shift, 3rd machine 150 units
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Properties of Matrices
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1) Addition Subtraction
4 shops A1, A2, A3, A4 each stocking items of
types p, q, r. The 4 columns represent the shops
and the 3 rows represent the item types.
A1
A2
A3
A4
Example-
Item of type p

• 3 4 5
• 4 5 6
• 5 6 7

Let A
Item of type q
Item of type r
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• 2 2 3
• 2 3 4
• 2 3 4 4

• 2 3 4
• 1 2 3
• 3 2 1 2

S
Let D1
where elements in D1 represent the number of
items of different types delivered at the
beginning of a week and matrix S represents the
sales during that week. Find
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• The number of items immediately after delivery of
  items.
• The number of items at the end of the week.
• The number of items needed to bring the stock of
  each item in each shop to 6.

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Solution.

• 2 3 4
• 1 2 3
• 3 2 1 2

• 3 4 5
• 4 5 6
• 5 6 7

D1
A
Delivery

• 5 7 9
• 5 7 9
• 7 7 7 9

• AD1

represents the number of items immediately after
the delivery of items.
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Solution.

• 2 2 3
• 2 3 4
• 2 3 4 4

• 2 3 4
• 1 2 3
• 3 2 1 2

• 3 4 5
• 4 5 6
• 5 6 7

S
A
D1
Sales

• 3 5 6
• 3 3 4 5
• 5 4 3 5

• (AD1) S

represents the number of items at the end of the
week
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• 3 5 6
• 3 3 4 5
• 5 4 3 5

(AD1) S

• We want each element in (AD1)-SD2 to be 6.

• 3 1 0
• 3 3 2 1
• 1 2 3 1

Let D2
Then (AD1)-SD2 is a matrix in which each
element is 6. So D2 represents the number of
items needed to bring the stock of each item in
each shop to 6.
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2) Multiplication
Example
In a town there are 20 colleges and 50
schools. Each school and college has 1 peon, 5
clerks, 1 cashier. Each college, in addition has
1 accountant and 1 head clerk. The monthly
salary of each of them is as follows peon Rs
1500, clerk Rs 2500, cashier Rs 3000, accountant
Rs 3500, head clerk Rs 4000.
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• Using matrix notation find,
• The total number of posts of each kind in schools
  and colleges taken together.
• The total monthly salary bill of each school and
  college separately.
• The total monthly salary bill of all the schools
  and colleges taken together.

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• Solution
• Consider the row matrix of order 1 x 2

This represents number of colleges and schools in
that order.
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Let
Staff
where columns represents number of peons, clerks,
cashier, accountants, head clerks while rows
represents colleges and schools in that order.
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1 5 1 1 1 1 5 1 0 0
S
(Remember row by column multiplication)
Then IS where the first element represents
total number of peons, second represents total
number of clerks, third represents total number
of cashiers, fourth represents total number of
accountants and fifth represents total number
of head clerks.
70 350 70 20 20
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• Let

where column matrix represents monthly salary of
peon, clerk, cashier, accountant and head clerk
in that order.
(Remember row by column multiplication)
1500 12 500 3000 3500 4000 1500 12 500
3000 0 0
Then SP
24 500 17 000
Thus total monthly salary bill of each college is
Rs 24 500 and of each school is Rs 17 000.

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24 500 17 000
SP

• Consider
• I(SP) (24 500 x 20 50 x 17 000)
• (490 000 850 000)
• (1 340 000)
• which gives the total monthly salary bill of
  schools and colleges.

Note that this is the same as (IS)P