Discuss Infinite Dimensional And Finite Dimensional Vector Spaces.

Finite Dimensional: A vector space is finite-dimensional if it has a basis with only finitely many vectors. (One reason for sticking to finite-dimensional spaces is so that the representation of a vector with respect to a basis is a finitely-tall vector, and so can be easily written.) We shall take the term "vector space" to mean "finite-dimensional vector space". Other spaces are interesting and important, but they lie outside of our scope.

Infinite Dimensional: The process of extending the algebraic and geometrical methods of linear algebra from matrices to differential or integral operators consists of going from a finite dimensional vector space, typically Rⁿ , to an infinite dimensional vector space, typically a function space. However, a vector space of functions has certain idiosyncrasies precisely because its dimension is infinite. These peculiarities are so important that we must develop the framework in which they arise.

The Row Echelon Form

A rectangular matrix is in row echelon form if it has the following three properties:

• All nonzero rows are above any rows of all zeros
• Each leading entry of a row is in a column to the right of the leading entry of the row above it
• All entries in a column below a leading entry are zeros
• The first nonzero number in any row is a 1

This matrix is in row echelon form:

Discuss Implicit Differentiation.

In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.  y can be given as a function of x implicitly rather than explicitly. When we have an equation R(x, y) = 0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x, y) with respect to x and y and then solve for dy/dx.
Example: Assume that y is a function of x . Find y' = dy/dx for x³ + y³ = 4 .
SOLUTION 1 : Begin with x3 + y3 = 4 . Differentiate both sides of the equation, getting

D ( x3 + y3 ) = D ( 4 ) ,

D ( x3 ) + D ( y3 ) = D ( 4 ) ,

(using the chain rule on D ( y3 ) .)

3x2 + 3y2 y' = 0 ,

so that (Now solve for y' .)

3y2 y' = - 3x2 , and

y' = -3x²/3y² = -x²/y² Ans.

 

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                           =     List and define the three basic types of operands     =
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There are three types of basic operands

1) Immediate - a constant
   An immediate operand is a constant expression such as a number, character or an    arithimetic expression.
   mov al,10          : al= 10
   mov bl, 'A'        : bl= 'A'
   mov cx, 'AB'       : cx= 'AB'
   mov dx, '123h'     : dx= '13h'

2) Register Operands
   Regester operands are eight or sixteen bit registers or 32 if using extended register.
   mov            ax,bx
   mov            al,bl


3 Memory Operands
  memory operands are specified either by the name of a variable or by a register that contains the address of a variable. A variable name implies the address of a variable and instructs the computer to reference the contents of memory at that address. Memory references have the following syntax:segment:offset(base, index, scale)



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                         =    Explain data addressing modes to from assembly programming      =
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Register(or Register Direct):
reg1 := reg2 * reg3;
This addressing mode does not have an effective address and is not considered to be an addressing mode on some computers.

Base plus offset, and variations:
reg := RAM[base + offset]
This is sometimes referred to as 'base plus displacement'

Immediate:
reg1 := reg2 + constant;

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                      =   Enlist and differentiate between program visible and invisible registers   .=
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Visible register:
Visible Registers are visible to programmers or in other words they can be accessed directly via instructions or the programs while Invisible Registers may be accessed indirectly by the instructions or may be accessed exclusively internally. The programming model of the 8086 through the Pentium II’s considered to be program visible because its registers are used during application programming and are specified by the instruc­tions.

invisible
- The program invisible registers are used to access and specify the address tables of global and local descriptor tables.

- Since these types of register cannot be accessed directly by a program they are called invisible registers.

- The Global Descriptor table register contains the limit and the base addresses for the descriptor table. The same applies for the Interrupt descriptor table register.

- Since the maximum length is limited to 64 Kbytes the limit of each descriptor table is limited to 16 bits.

- The GTDR is loaded with the address of the global descriptor table whenever a protected mode operation is required.



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                              =    Differentiate between selectors and descriptors      =
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Descriptors are a bit like real mode segments; they describe (as the name implies) a memory area in protected mode. A descriptor contains information about segment length, its base address and the attributes of it (i.e. type, access rights, ...). These descriptors are stored internally in a so-called descriptor table, which is basically an array of such descriptors. Selectors are roughly an index into this table. Because these 'segments' can be up to 4 GB in size, 32 bits aren't sufficient anymore to describe a single memory location like in real mode. 48 bits are now needed to do this, a 32 bit address and a 16 bit sized selector. The GO32 unit provides the tseginfo record to store such a pointer. But due to the fact that most of the time data is stored and accessed in the %ds selector, FPC assumes that all pointers point to a memory location of this selector. So a single pointer is still only 32 bits in size. This value represents the offset from the data segment base address to this memory location.

Discuss sorting; what are different sorting techniques:

sorting is ordering a list of objects. We can distinguish two types of sorting. If the number of objects is small enough to fits into the main memory, sorting is called internal sorting. If the number of objects is so large that some of them reside on external storage during the sort, it is called external sorting.
type of sorting:

O(n2) algorithms
Bubble Sort


The algorithm works by comparing each item in the list with the item next to it, and swapping them if required. In other words, the largest element has bubbled to the top of the array. The algorithm repeats this process until it makes a pass all the way through the list without swapping any items.

void bubbleSort(int ar[])
{
   for (int i = (ar.length - 1); i >= 0; i--)
   {
      for (int j = 1; j = i; j++)
      {
         if (ar[j-1] > ar[j])
         {
              int temp = ar[j-1];
              ar[j-1] = ar[j];
              ar[j] = temp;
   } } } }

Example. Here is one step of the algorithm. The largest element - 7 - is bubbled to the top:

    7, 5, 2, 4, 3, 9
    5, 7, 2, 4, 3, 9
    5, 2, 7, 4, 3, 9
    5, 2, 4, 7, 3, 9
    5, 2, 4, 3, 7, 9
    5, 2, 4, 3, 7, 9


Selection Sort
The algorithm works by selecting the smallest unsorted item and then swapping it with the item in the next position to be filled.

The selection sort works as follows: you look through the entire array for the smallest element, once you find it you swap it (the smallest element) with the first element of the array. Then you look for the smallest element in the remaining array (an array without the first element) and swap it with the second element. Then you look for the smallest element in the remaining array (an array without first and second elements) and swap it with the third element, and so on. Here is an example,

void selectionSort(int[] ar){
   for (int i = 0; i ‹ ar.length-1; i++)
   {
      int min = i;
      for (int j = i+1; j ‹ ar.length; j++)
            if (ar[j] ‹ ar[min]) min = j;
      int temp = ar[i];
      ar[i] = ar[min];
      ar[min] = temp;
} }

Prove that ~[r ∨ (q ∧ (~r →~p))] ≡ ~r ∧ (p∨ ~q) by using a
series of logical equivalences.

Take RHS ~r ∧ (p ∨~q)

≡ ~r ∧ (~q ∨ p) Commutative law applied

≡ (~r ∧¬q) ∨ (~r ∧ p) Distributive law applied

≡ (~r ∧~q) ∨ ((~r ∧ ~r) ∧ p) Idempotent law applied≡ (~r ∧¬q) ∨ (~r ∧ (~r ∧ p)) Associative law applied

≡ ~r ∧ (~q ∨ (~r ∧ p)) Distributive law applied

≡ ~r ∧ (~q ∨ ~ (r ∨ ~p)) De Morgan’s law & double negation applied

≡ ~r ∧ ~ (q ∧ (r ∨~p)) De Morgan’s law applied

≡ ~r ∧ ~ (q ∧ (~~r ∨ ~p)) Double negation law applied

≡ ~r ∧ ~ (q ∧ (~r → ~p)) Conditional rewritten as disjunction applied≡ ¬[r ∨ (q ∧ (¬r → ¬p))] De Morgan’s law applied

Proved that ~ [r ∨ (q ∧ (~r →~p))] ≡ ~r ∧ (p∨ ~q)

Discuss about the LU Factorizations.

In linear algebra (Lu-factorization)  LU decomposition factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. Let A be a square matrix. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U,

A=LU