elementaryrowoperations

elementaryrowoperations - Elementary Row Operations Math...

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Elementary Row Operations Math 108A: April 7-9, 2010 John Douglas Moore A. Homogeneous linear systems Linear algebra is the theory behind solving systems of linear equations, such as a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = 0 , a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = 0 , · · · · · · · · a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = 0 . (1) Here the a ij ’s are known elements of the field F , and we are solving for the unknown elements x 1 , . . . , x n in F . Our goal is to describe the space of solutions W = { ( x 1 , x 2 , . . . , x n ) F n : ( x 1 , x 2 , . . . , x n ) satisfies (1) } as simply as possible. You will note that W is a linear subspace of F n . In fact, it turns out that any linear subspace of F n is a solution set to a homogeneous linear system of equations just like (1). Our strategy is to simplify the system by means of the elementary operations on equations : 1. Interchange two equations. 2. Multiply an equation by a nonzero constant c . 3. Add a constant multiple of one equation to another. All of these operations are reversible and each leads to a new system with exactly the same solution set W . We want to choose these operations judiciously, so that we put the system into the simplest possible form. From properties of matrix multiplication that you learned in Math 3C, you realize that this system of linear equations can be written in terms of its coeffi- cient matrix A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n · · · · · · a m 1 a m 2 · · · a mn and the vector x = x 1 x 2 · x n . as A x = 0 . Thus the solution set W can be expressed more simply as W = { x F n : A x = 0 } . 1
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Each elementary operation on the linear system (1) corresponds to elementary row operations on its coefficient matrix A . Those elementary row operations are: 1. Interchange two rows. 2. Multiply a row by a nonzero constant c . 3. Add a constant multiple of one row to another. Each of these operations is reversible and leaves the solutions to the matrix equation A x = 0 unchanged. Our goal is to use these operations to replace A by a matrix that is in row-reduced echelon form. By definition, the matrix A is in row-reduced echelon form if it has the following properties: 1. The first nonzero entry in any row is a one.
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