Presentation Linear Algebra.pptx easy pp

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1. Linear Algebra Basic definition and a pplication

2. Group Members Amina Aslam FA18-EPE-007 Momina Munir FA18-EPE-010 Hafsa Maqasad FA18-EPE-043 Memona Nawaz FA18-Epe-044

3. Definition : Linear algebra   is the branch of mathematics concerning  linear  equations such as:  linear  maps such as: and their representations in vector spaces and through matrices.  Linear algebra  is central to almost all areas of mathematics. Linear equation : linear equation  is an algebraic  equation  in which each term has an exponent of one and the graphing of the  equation  results in a straight line . An  example  of  linear equation  is y=mx + b.

4. System of Linear E quations : A  system of linear Equations  is just a set of two or more  linear equations . In two variables (x and y) , the graph of a  system  of two  equations  is a pair of lines in the plane. EXAMPLE :   1) 3 x +2 y =16 2) 4 x+3y =− 2 7x+y=19 8x − 2y=12 A  solution  of a linear system is an assignment of values to the variables  x 1 ,  x 2 , ...,  x n  such that each of the equations is satisfied. The  set  of all possible solutions is called the  solution set . A linear system may behave in any one of three possible ways: The system has  infinitely many solutions . The system has a single  unique solution . The system has  no solution      

5. Consistency Consistent Equation A  linear  or  nonlinear   system of equations  is called  consistent  if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when  substituted  into each of the equations, they make each equation hold true as an  identity . EXPAMPLE:  2 = 1, or  x 3  +  y 3  = 5  x 3  +  y 3  = 6 Inconsistent Equation a linear or non linear equation system is called  inconsistent  if there is no set of values for the unknowns that satisfies all of the equations . EXPAMPLE: X+Y+Z=3 X+Y+Z=4

6. Criteria For Consistency Linear systems : A linear system is consistent  if and only if  its  coefficient matrix  has the same  rank  as does its  augmented matrix . If coefficient matric rank in not equal to agumanted matric than it is in consistent

7. Applications Of L inear Algebra : Linear Algebra most apparently uses by Electrical engineers. Various electrical circuits solution like Kirchhoff's laws , ohms law.

8. The Wheatstone Bridge : A simple circuit for the precise measurements of resistors . Invented by Samuel hunter Christy in 1883 , was named after sir chales Wheatstone who found and popularized the arrangement in 1843. Consist of an electrical source and a galvanometer get connects to parallel branches 4 resistors . 3 of which are known. Example: We absorb a circuit that has a 5v power supply with different loops and resistors. SOLUTION :

9. We start with general equation Where V is = Voltage I=current Rn=Total resistance of the path for the given current in.

10. Next we want to look at each loop and set up an equation. We can put these equation in an augmented matrix Row reduced this matrix we get From this we can determine what the current through I1, I2 & I3 are .

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