Linear Algebra Essay

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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 and 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 mass known to be 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. 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(C¬7H8 ) + 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. 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

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