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Linear Equations and Matrices

CHAPTER 3. Linear Equations and Matrices In this chapter we introduce Matrices via the theory of simultaneous Linear Equations . This method has the advantage of leading in a natural way to the concept of the reduced row-echelon form of a matrix. In addition, we will for- mulate some of the basic results dealing with the existence and uniqueness of systems of Linear Equations . In Chapter 5 we will arrive at the same matrix algebra from the viewpoint of Linear transformations. SYSTEMS OF Linear Equations . Let a , .. , a , y be elements of a field F, and let x , .. , x be unknowns (also called variables or indeterminates). Then an equation of the form a x + ~ ~ ~ + a x = y is called a Linear equation in n unknowns (over F).

115 C H A P T E R 3 Linear Equations and Matrices In this chapter we introduce matrices via the theory of simultaneous linear equations. This method has the advantage of leading in a natural way to the

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