Chapter 1.1 SystemsLinearEquations

Outline linear systems matrices gaussian elimination

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Unformatted text preview: + λa1n xn = λb1 . Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Elementary Row Operations If we apply any one of the following three operations to a system of linear equations: Interchange two equations. Multiply an equation by a nonzero constant. Add a multiple of one equation to another equation. then we obtain an equivalent system. These are called elementary row operations. We can systematically eliminate variables from a system of linear equations via elementary row operations to arrive at an equivalent system whose solutions are easy to see. Outline Linear Systems Matrices Gaussian Elimination Examples Example Solve the system of linear equations using elementary row operations: x + 2y + z = 3 3x − y − 3z = −1 2x + 3 y + z = 4 Homogeneous Systems Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Consistent and Inconsistent Systems A system is consistent if it has at least one solution. A system with no solutions is said to be inconsistent. y y x y x Theorem A system of m linear equations in n unknowns has exactly one solution, no solutions, or infinitely many solutions. x Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Example Determine if the systems are consistent or inconsistent. Solve the system if it is consistent. 1 x 2x 4x + 2y −y + 3y +z +z + 3z =1 =2 = −3 2 x 2x 4x + 2y −y + 3y +z +z + 3z =1 =2 =4 Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Matrices and Linear Systems Consider the linear system: a11 x1 + a21 x1 + . . . a12 x2 + · · · + a22 x2 + · · · + . .. . . . am1 x1 + am2 x2 + · · · a1n xn = a2n xn = . . . b1 b2 . . . (3) + amn xn = bm If we align vertically the like variables in the system, only the coefficients are necessary to perform the row operations. This implies that we can turn the system into a rectangular array of numbers, called a matrix, that contains all of the critical information about the s...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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