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Matrix Algebra bca 1st wbut

Matrix Algebra
We review here some of the basic denitions and elementary algebraic operations on matrices.
There are many applications as well as much interesting theory revolving around these con-
cepts, which we encourage you to explore after reviewing this tutorial.
A matrix is simply a retangular array of numbers. For example,
A =
266664
a11 a12


a1n
a21 a22


a2n
...
...
. . .
...
am1 am2


amn
377775
is a mn matrix (m rows, n columns), where the entry in the ith row and jth column is aij .
We often write A = [aij ].
Some Terminology
For an n  n square matrix A, the elements a11; a22; : : : ; ann form the main diagonal of
the matrix. The sum
n Pk=1
akk of the elements on the main diagonal of A is called the trace
of A.
The matrix AT = [aji] formed by interchanging the rows and columns of A is called the
transpose of A. If AT = A, the matrix A is symmetric.
Example
Let B = " 6 9
􀀀4 􀀀6 #. The trace of B is 6 + (􀀀6) = 0.
The transpose of B is BT = " 6 􀀀4
9 􀀀6 #.
Addition and Subtraction of Matrices
To add or subtract two matrices of the same size, simply add or subtract corresponding
entries. That is, if B = [bij ] and C = [cij ],
B + C = [bij + cij ] and B 􀀀 C = [bij 􀀀 cij ]:

Mathematics syllabus 1st sem bca

Mathematics
Code: BM 101
CREDITS: 4
Algebra: Sets, Union, intersection, complement, mapping, notion of group, ring, field with simple examples; Polynomials, division algorithm, fundamental theorem of classical algebra (without proof), Descartes rule of sign and their application, relation between roots and coefficients, symmetric function of roots, transformation of polynomial equations, Cardan’s solution of cubic equation.
Matrices, addition and multiplication of matrices, inverse matrix, solution of linear equations in three variables by Cramer’s rule, solution of three line linear equations by matrix inversion method.
Differential calculus: Limits of function and continuity, fundamental properties of continuous functions (without proof), geometric meaning of derivative and differential, rules of differentiation, successive differentiation, Rolle’s theorem, mean value theorem, Taylor’s and Maclaurin’s theorems with Cauchy’s and Lagrange’s forms of reminder, Taylor’s series, function of several variables, partial derivatives, total differential, Euler’s theorem on homogeneous functions of two variables.
Introduction to:
Application to plane curves.
Integral calculus: Rules of integration of indefinite integrals, solution of definite integrals and their elementary properties, idea of improper integrals.
Dimensional geometry: Transformation of rectangular axes, invariants, general equation of second degree – reduction to standard forms and classification, plane polar equation of a straight line, circle and conic.

Introduction To Programming BCA WBUT 1ST SEM

Introduction To Programming

Code: BCA 103

CREDITS: 4

Problem analysis, need for programmed languages, introduction to algorithms, algorithmic representations, Pseudocodes flow charts and decision tables, structured programming and modular programming.

Overview of C.

Constant, variables, data, types and size, variable declaration, operators and expressions, type conversion, conditional expression, special operators, precedence rules. Decision making, looping and control structures. Function, recursion, arrays, pointers, structures and unions,

Managing input/output operations, formatted I/O, standard library/user-defined functions, file management in C.

Handling of character strings, Dynamic memory allocations, linked list, Pre-processor.

Developing different C Programs.

Digital Electronics Syllabus BCA WBUT 1ST SEM

Digital Electronics Syllabus

Code: BCA101

CREDITS: 4

Data and number representation- binary-complement representation BCD-ASCII, conversion of numbers form one system to the other, 2’s complement representation, binary arithmetic

Logic gates, basic logic operations, truth tables, Boolean expression, simplification

Combination circuits, adders, Multiplexer, Sequential circuits, flip-flops, Registers, counter(Async &Sync)

Memory circuits, ROM, PROM, EPROM and dynamic RAM, Digital Components

Introduction

Introduction

Digital and Analog Signals

Signals carry information and are defined as any physical quantity that varies with time, space, or any other independent variable. For example, a sine wave whose amplitude varies with respect to time or the motion of a particle with respect to space can be considered as signals. A system can be defined as a physical device that performs an operation on a signal. For example, an amplifier is used to amplify the input signal amplitude. In this case, the amplifier performs some operation(s) on the signal, which has the effect of increasing the amplitude of the desired information-bearing signal.

Signals can be categorized in various ways; for example discrete and continuous time domains. Discrete-time signals are defined only on a discrete set of times. Continuous-time signals are often referred to as continuous signals even when the signal functions are not continuous; an example is a square-wave signal.

Figure 1a: Analog Signal

Figure 1b: Digital Signal

Another category of signals is discrete-valued and continuous-valued or otherwise known as digital and analog signals. Digital signals are discrete-valued and analog signals are continuous electrical signals that

vary in time as shown in Figure 1 (a) and (b). Analog devices and systems process signals whose voltages or other quantities vary in a continuous manner. They can take on any value across a continuous range of voltage, current, or other metric. The analog signals can have an infinite number of values. Analog systems can be called wave systems. They have a value that changes steadily over time and can have any one of an infinite set of values in a range. Analog signals represent some physical quantity and they can be a model of the real quantity. Most of the time, the variations corresponds to that of the non-electric (original) signal. For example, the telephone transmitter converts the sounds into an electrical voltage signal. The intensity of the voice causes electric current variations. Therefore, the two are analogous hence the name analog. At the receiving end, the signal is reproduced in the same proportion. Hence the electric current is a model and is an electrical representation of one's voice.

Not all analog signals vary as smoothly as the waveform shown in Fig 1(a). Digital signals are non-continuous, they change in individual steps. They consist of pulses or digits with discrete levels or values. The value of each pulse is constant, but there is an abrupt change from one digit to the next. Digital signals have two amplitude levels. The value of which are specified as one of two possibilities such as 1 or 0, HIGH or LOW , TRUE or FALSE and so on. In reality, the values are anywhere within specific ranges and we define values within a given range.

A digital system is the one that handles only discrete values or signals. Any set that is restricted to a finite number of elements contains discrete information. The word digital describes any system based on discontinuous data or events. Digital is the method of storing, processing and transmitting information through the use of distinct electronic pulses that represent the binary digits 0 and 1. Examples of discrete sets are the 10 decimal digits, the 26 letters of the alphabet etc. A digital system would be to flick the light switch on and off. There's no 'in between' values.

Advantages of digital signals

The usual advantages of digital circuits when compared to analog circuits are:

Noise Margin (resistance to noise/robustness): Digital circuits are less affected by noise. If the noise is below a certain level (the noise margin), a digital circuit behaves as if there was no noise at all. The stream of bits can be reconstructed into a perfect replica of the original source. However, if the noise exceeds this level, the digital circuit cannot give correct results.

Error Correction and Detection: Digital signals can be regenerated to achieve lossless data transmission, within certain limits. Analog signal transmission and processing, by contrast, always introduces noise.

Easily Programmable: Digital systems interface well with computers and are easy to control with software. It is often possible to add new features to a digital system without changing hardware, and to do this remotely, just by uploading new software. Design errors or bugs can be worked-around with a software upgrade, after the product is in customer hands. A digital system is often preferred because of (re-)programmability and ease of upgrading without requiring hardware changes. mywbut.com

Cheap Electronic Circuits: More digital circuitry can be fabricated per square millimeter of integrated-circuit material. Information storage can be much easier in digital systems than in analog ones. In particular, the great noise-immunity of digital systems makes it possible to store data and retrieve it later without degradation. In an analog system, aging and wear and tear will degrade the information in storage, but in a digital system, as long as the wear and tear is below a certain level, the information can be recovered perfectly. Theoretically, there is no data-loss when copying digital data. This is a great advantage over analog systems, which faithfully reproduce every bit of noise that makes its way into the signal.

Disadvantages - The world in which we live is analog, and signals from this world such as light, temperature, sound, electrical conductivity, electric and magnetic fields, and phenomena such as the flow of time, are for most practical purposes continuous and thus analog quantities rather than discrete digital ones. For a digital system to do useful things in the real world, translation from the continuous realm to the discrete digital realm must occur, resulting in quantization errors. This problem can usually be mitigated by designing the system to store enough digital data to represent the signal to the desired degree of fidelity. The Nyquist-Shannon sampling theorem provides an important guideline as to how much digital data is needed to accurately portray a given analog signal.

Digital systems can be fragile, in that if a single piece of digital data is lost or misinterpreted, the meaning of large blocks of related data can completely change. This problem can be diminished by designing the digital system for robustness. For example, a parity bit or other error-detecting or error-correcting code can be inserted into the signal path so that minor data corruptions can be detected and possibly corrected.

Digital circuits use more energy than analog circuits to accomplish the same calculations and signal processing tasks, thus producing more heat as well. In portable or battery-powered systems this can be a major limiting factor.

Digital circuits are made from analog components, and care has to be taken to all noise and timing margins, to parasitic inductances and capacitances, to proper filtering of power and ground connections, to electromagnetic coupling amongst data lines. Inattention to these can cause problems such as "glitches", pulses do not reach valid switching (threshold) voltages, or unexpected ("undecoded") combinations of logic states.

A corollary of the fact that digital circuits are made from analog components is the fact that digital circuits are slower to perform calculations than analog circuits that occupy a similar amount of physical space and consume the same amount of power. However, the digital circuit will perform the calculation with much better repeatability, due to the high noise immunity of digital circuitry.

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Number System

Introduction

Number systems provide the basis for all operations in information processing systems. In a number system the information is divided into a group of symbols; for example, 26 English letters, 10 decimal digits etc. In conventional arithmetic, a number system based upon ten units (0 to 9) is used. However, arithmetic and logic circuits used in computers and other digital systems operate with only 0's and 1's because it is very difficult to design circuits that require ten distinct states. The number system with the basic symbols 0 and 1 is called binary. ie. A binary system uses just two discrete values. The binary digit (either 0 or 1) is called a bit.

A group of bits which is used to represent the discrete elements of information is a symbol. The mapping of symbols to a binary value is known a binary code. This mapping must be unique. For example, the decimal digits 0 through 9 are represented in a digital system with a code of four bits. Thus a digital system is a system that manipulates discrete elements of information that is represented internally in binary form.

Decimal Numbers

The invention of decimal number system has been the most important factor in the development of science and technology. The decimal number system uses positional number representation, which means that the value of each digit is determined by its position in a number.

The base, also called the radix of a number system is the number of symbols that the system contains. The decimal system has ten symbols: 0,1,2,3,4,5,6,7,8,9. In other words, it has a base of 10. Each position in the decimal system is 10 times more significant than the previous position. The numeric value of a decimal number is determined by multiplying each digit of the number by the value of the position in which the digit appears and then adding the products. Thus the number 2734 is interpreted as

Here 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).

In general in a number system with a base or radix r, the digits used are from 0 to r-1 and the number can be represented as mywbut.com

1

Equation (1) is for all integers and for the fractions (numbers between 0 and 1), the following equation holds.

Thus for decimal fraction 0.7123

Binary Numbers

The binary number has a radix of 2. As r = 2, only two digits are needed, and these are 0 and 1. Like the decimal system, binary is a positional system, except that each bit position corresponds to a power of 2 instead of a power of 10. In digital systems, the binary number system and other number systems closely related to it are used almost exclusively. Hence, digital systems often provide conversion between decimal and binary numbers. The decimal value of a binary number can be formed by multiplying each power of 2 by either 1 or 0 followed by adding the values together.

Example: The decimal equivalent of the binary number 101010.

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In binary r bits can represent symbols. e.g. 3 bits can represent up to 8 symbols, 4 bits for 16 symbols etc. For N symbols to be represented, the minimum number of bits required is the lowest integer 'r'' that satisfies the relationship.

E.g. if N = 26, minimum r is 5 since.

Octal Numbers

Digital systems operate only on binary numbers. Since binary numbers are often very long, two shorthand notations, octal and hexadecimal, are used for representing large binary numbers. Octal systems use a base or radix of 8. Thus it has digits from 0 to 7 (r-1). As in the decimal and binary systems, the positional valued of each digit in a sequence of numbers is fixed. Each position in an octal number is a power of 8, and each position is 8 times more significant than the previous position.

Example: The decimal equivalent of the octal number 15.2.

Hexadecimal Numbers

The hexadecimal numbering system has a base of 16. There are 16 symbols. The decimal digits 0 to 9 are used as the first ten digits as in the decimal system, followed by the letters A, B, C, D, E and F, which represent the values 10, 11,12,13,14 and 15 respectively. Table 1 shows the relationship between decimal, binary, octal and hexadecimal number systems.

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Decimal

Binary

Octal

Hexadecimal

Decimal

Binary

Octal

Hexadecimal

Hexadecimal numbers are often used in describing the data in computer memory. A computer memory stores a large number of words, each of which is a standard size collection of bits. An 8-bit word is known as a Byte. A hexadecimal digit may be considered as half of a byte. Two hexadecimal digits constitute one byte, the rightmost 4 bits corresponding to half a byte, and the leftmost 4 bits corresponding to the other half of the byte. Often a half-byte is called nibble.

If "word" size is n bits there are 2n possible bit patterns so only 2n possible distinct numbers can be represented. It implies that all possible numbers cannot be represent and some of these bit patterns (half?) to represent negative numbers. The negative numbers are generally represented with sign magnitude i.e. reserve one bit for the sign and the rest of bits are interpreted directly as the number. For example in a 4 bit system, 0000 to 0111 can be used to positive numbers from +0 to +2n-1 and represent 1000 to 1111 can be used for negative numbers from -0 to -2n-1. The two possible zero's redundant and also it can be seen that such representations are arithmetically costly.

Another way to represent negative numbers are by radix and radix-1 complement (also called r's and (r-1)'s). For example -k is represented as Rn -k. In the case of base 10 and corresponding 10's complement with n=2, 0 to 99 are the possible numbers. In such a system, 0 to 49 is reserved for positive numbers and 50 to 99 are for positive numbers.

Examples: mywbut.com

4

+3 = +3 -3 = 10 2 -3 = 97

2's complement is a special case of complement representation. The negative number -k is equal to 2 n -k. In 4 bits system, positive numbers 0 to 2n-1 is represented by 0000 to 0111 and negative numbers -2n-1 to -1 is represented by 1000 to 1111. Such a representation has only one zero and arithmetic is easier. To negate a number complement all bits and add 1

Example:

119 10 = 01110111 2

Complementing bits will result

10001000 +1 add 1 10001001 That is 10001001 2 = - 119 10

Properties of Two's Complement Numbers

1. X plus the complement of X equals 0.

2. There is one unique 0.

3. Positive numbers have 0 as their leading bit (MSB); while negatives have 1 as their MSB.

4. The range for an n-bit binary number in 2's complement representation is from -2 (n-1) to 2 (n-1) - 1

5. The complement of the complement of a number is the original number.

6. Subtraction is done by addition to the 2's complement of the number.

Value of Two's Complement Numbers For an n-bit 2's complement number the weights of the bits is the same as for unsigned numbers except of the MSB. For the MSB or sign bit, the weight is -2 n-1. The value of the n-bit 2's complement number is given by:

A 2's-complement = (a n-1) x (-2 n-1) + (a n-2 ) x (2 n-1 ) + ... (a 1 ) x (2 1 ) + a 0

For example, the value of the 4-bit 2's complement number 1011 is given by:

= 1 x -2 3 + 0 x 2 2 + 1 x 2 1 + 1 = -8 + 0 + 2 + 1 = -5

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An n-bit 2's complement number can converted to an m-bit number where m>n by appending m-n copies of the sign bit to the left of the number. This process is called sign extension. Example: To convert the 4-bit 2's complement number 1011 to an 8-bit representation, the sign bit (here = 1) must be extended by appending four 1's to left of the number:

1011 4-bit 2's-complement = 11111011 8-bit 2's-complement

To verify that the value of the 8-bit number is still -5; value of 8-bit number

= -27 + 26 + 25 + 24 + 23 +2 +1

= -128 + 64 + 32 + 16 +8 +2+1

= -128 + 123 = -5

Similar to decimal number addition, two binary numbers are added by adding each pair of bits together with carry propagation. An addition example is illustrated below:

X 190 Y 141 X + Y 331

Similar to addition, two binary numbers are subtracted by subtracting each pair of bits together with borrowing, where needed. For example:

X 229 Y 46 X - Y 183

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Two' complement addition/subtraction example

Overflow occurs if signs (MSBs) of both operands are the same and the sign of the result is different. Overflow can also be detected if the carry in the sign position is different from the carry out of the sign position. Ignore carry out from MSB.

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Number Base Conversion

This section describes the conversion of numbers from one number system to another. Radix Divide and Multiply Method is generally used for conversion. There is a general procedure for the operation of converting a decimal number to a number in base r. If the number includes a radix point, it is necessary to separate the number into an integer part and a fraction part, since each part must be converted differently. The conversion of a decimal integer to a number in base r is done by dividing the number and all successive quotients by r and accumulating the remainders. The conversion of a decimal fraction is done by repeated multiplication by r and the integers are accumulated instead of remainders.

Integer part - repeated divisions by r yield LSD to MSD

Fractional part - repeated multiplications by r yield MSD to LSD

Example: Conversion of decimal 23 to binary is by divide decimal value by 2 (the base) until the value is 0

The answer is 23 10 = 10111 2

Divide number by 2; keep track of remainder; repeat with dividend equal to quotient until zero; first remainder is binary LSB and last is MSB.

The conversion from decimal integers to any base-r system is similar to this above example, except that division is done by r instead of 2.

Example: Convert (0.7854) 10 to binary.

0.7854 x 2 = 1.5708; a -1 = 1

0.5708 x 2 = 1.1416; a -2 = 1

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0.1416 x 2 = 0.2832; a -3 = 0

0.2832 x 2 = 0.5664; a -4 = 0

The answer is (0.7854) 10 = (0.1100) 2

Multiply fraction by two; keep track of integer part; repeat with multiplier equal to product fraction; first integer is MSB , last is the LSB; conversion may not be exact; a repeated fraction. The conversion from decimal fraction to any base-r system is similar to this above example, except the multiplication is done by r instead of 2.

The conversion of decimal numbers with both integer and fraction parts is done by converting the integer and the fraction separately and then combining the two answers.

Thus (23.7854) 10 = (10111. 1100) 2

For converting a binary number to octal, the following two step procedure can be used.

1. Group the number of bits into 3's starting at least significant symbol. If the number of bits is not evenly divisible by 3, then add 0's at the most significant end.

2. Write the corresponding 1 octal digit for each group

Examples:

Similarly for converting a binary number to hex, the following two step procedure can be used.

1. Group the number of bits into 4's starting at least significant symbol. If the number of bits is not evenly divisible by 4, then add 0's at the most significant end.

2. Write the corresponding 1 hex digit for each group

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Examples:

The hex to binary conversion is very simple; just write down the 4 bit binary code for each hexadecimal digit

Example:

Similarly for octal to binary conversion, write down the 8 bit binary code for each octal digit.

The hex to octal conversion can be carried out in 2 steps; first the hex to binary followed by the binary to octal. Similarly, decimal to hex conversion is completed in 2 steps; first the decimal to binary and from binary to hex as described above.

Business Systems and Applications BCA WBUT 1ST SEM

Business Systems and Applications
Code: BCA102
CREDITS: 4
Use of computers for managerial applications, Technology issues and data processing in organisations, Introduction to Information Systems, shift in Information system thinking, latest trends in Information Technology
Computer Based Information Systems- office automation systems. Decision making and MIS, transaction processing systems.
Decision support system, Group Decision Support, Executive Information systems, DSS generator
Introduction to:
Artificial Intelligence Based Systems, End user computing, Distributed data processing.
Deciding on IS architecture, IT leadership & IS strategic planning.
Introduction to:
IS strategy and effects of IT on competition.
Introduction to:
ERP, re-engineering work processes for IT applications, Business Process Redesign
Knowledge engineering and data warehouse.

PC SOFTWARE WBUT 1ST SEM

PC Software
Code: BCA 104
CREDITS: 4
Introduction MS Windows(Windows ’98 Second Edition)
Desktop, creation of folders and shortcuts, features of Windows explorer
Familiarisation and using MS packages – Word, Excel, PowerPoint, basic skills in using these tools.(Version MS-Office’2000)