Linear Algebra Discipline - Term Paper Example

Linear Algebra INTRODUCTION Linear Algebra is a mathematical discipline related to linear equations, linear transformations, vectors and vector spaces, and matrices. Unlike other areas of mathematics, which are frequented by new and unsolved problems, Linear Algebra has firm sets of rules which are well defined and understood. Its significance is most appreciated by observing its applications in mathematical physics, modern algebra and coding theory. LINEAR EQUATIONS Systems of linear equations are common in physics, chemistry and mathematics. The two examples below give a sense of how they are used. The first example is from Physics. Suppose that we are given three objects, one with a known mass of 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances. Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations (Hefferon). 40h + 15c = 100 25c = 50 + 50h The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene C7H8 and nitric acid HNO3 to produce trinitrotoluene C7H5O6N3 along with the byproduct water (conditions have to be controlled very well, indeed-trinitrotoluene is better known as TNT). In what proportion should those components be mixed x(C7H8 ) + y(HNO3) z(C7H5O6N3) + w(H2O) The number of atoms of each element present before the reaction must equal the number present afterward. Applying that principle to the elements C, H, N, and O in turn gives this system (Hefferon). 7x = 7z (counting 'C' on both sides and equating) 8x + 1y = 5z + 2w (counting 'H' on both sides and equating) 1y = 3z (counting 'N' on both sides and equating) 3y = 6z + 1w (counting 'O' on both sides and equating) Our next example is about solving a riddle. There are two groups of people X and Y having certain number of persons in each group. If a person from X leaves to join Y then Y becomes double of X. If a person leaves Y to join X then they both become equal to each other. How many persons are there in each group Y+1=2(X-1) or Y=2X-3 .(i) Y-1=X+1 or Y=X+2 ..(ii) Therefore X+2=2X-3 or X=5 Substituting the value of X in either of the two equations above gives Y=7. This third example was to demonstrate that to finish the first two requires solving of a system of equations. All the equations involve only the first power of variables. A linear equation in n unknowns x1, x2, x3, , xn is an equation of the form a1x1 + a2x2 + a3x3 + anxn = b, where a1, a2, a3, , an and b are real and given number and x1, x2, x3, , xn are the unknowns. A system of m linear equations in n unknowns thus becomes a family of equations of the form: a11x1 + a12x2 + a13x3 + a1nxn = b1 a21x1 + a22x2 + a23x3 + a2nxn = b2 a31x1 + a32x2 + a33x3 + a3nxn = b3 . . . am1x1 + am2x2 + am3x3 + amnxn = bm The objective is to determine if such system of linear equations has a solution or not. That is to find out if there exist values of x1 to xn which when fed into these equation will simultaneously satisfy all the equations. If true then the system is said to be consistent or else it is inconsistent (Matthews). MATRICES The above system of equations can also be briefly written as, The matrix is called the coefficient matrix of the system of equations as it only has the coefficients of variables listed in it. If this matrix were also to include the constants involved in the equations then it would be called an augmented matrix of the system and would be written as, Three elementary row operations can be performed on matrices which do not affect the solution of linear equations. 1. Interchanging two rows 2. Multiplying a row by a non-zero number 3. Adding a multiple of one row to another We will try to solve the following equations with the help of a matrix and then applying any or all of the appropriate elementary row operations stated above. We have three equations, or a matrix of the form then multiply second row with 3 and add it to the first row, we get , this clearly gives y = 5 and thus x = -1. VECTORS & VECTOR SPACES In a slightly advanced stage Linear Algebra vectors are normally the starting point. Vectors are understood to be those quantities which have a magnitude and direction as well. Vectors let physical applications make the best use of them. For example, when a solid object, free to move, simultaneously experiences two different forces the net result is like it has been moved by some single force. To visualize this, the two forces, namely 'A' and 'B' are drawn as arrows in Figure 2. The direction of arrows represents the direction of force and the lengths of arrows depict their magnitude. The net force which is the result of adding the two forces 'A' and 'B' is shown in the diagram as 'A + B' or the sum of 'A' and 'B'. Vectors are added by initially placing their tails at the origin of an xy coordinate system with their lengths and directions remaining the same as they were. In the next step, coordinates of the arrowheads are added to each other, x to x and y to y, and a resultant vector is obtained. As shown in Figure 3, the resultant vector forms a diagonal of parallelogram shaped by the original vectors. LINEAR TRANSFORMATION & VECTOR SPACES Vectors when added to another vector provide a resultant vector with direction and magnitude of its own. Vectors can also be multiplied by scalar quantities resulting in a vector which maintains its direction but its length represents the product of its original magnitude and the scalar quantity used to multiply. A set of such vectors is called a vector space. The different methods of sending or mapping one vector to another are called linear transformation. If a vector v is multiplied by a constant x then it amounts to saying that this has transformed vector v to a vector which is x times the vector v, or xv. The direction of transformed vector xv remains unchanged but its magnitude is now x times itself. If v is a set of vectors then multiplying this set by a constant x will transform all the vectors by x times. APPLICATIONS Out of the numerous applications of Linear Algebra some are listed below. 1. Coding Theory 2. Cryptography 3. Elimination Theory 4. Genetics 5. Image compression 6. Linear Programming Bibliography Hefferon, Jim. Linear Algebra. Colchester, 20 May 2006. "Linear Algebra." Ultimate Reference Suite. Chicago: Encyclopdia Britannica, 2008. Matthews, K. R. Elementary Linear Algebra. 2nd Online Edition. Queensland, December 1998. Microsoft Corporation. Encarta Book of Quotations. Bloomsbury Publishing PLc., 1999. -. Encarta World English Dictionary. Bloomsbur Publishing Plc, 1999. -. "Linear Algebra." Microsoft Encart 2008. Redmond: Microsoft Corporation, 2007. Read More