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Chapter 5 Slides

Chapter 5 Slides - Systems of Linear Algebraic Equations...

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Systems of Linear Algebraic Equations Example Solve the system x + 2 y - 5 z = - 1 - 3 x - 9 y + 21 z = 0 x + 6 y - 11 z = 1 -→ x + 2 y - 5 z = - 1 y - 2 z = 1 z = - 1 Solution set: x = - 4 , y = - 1 , z = - 1 1

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The Elementary Operations The operations that produce equiv- alent systems are called elementary operations . 1. Multiply an equation by a nonzero number. 2. Interchange two equations. 3. Multiply an equation by a num- ber and add it to another equation. 2
Example Solve the system x 1 - 2 x 2 + x 3 - x 4 = - 2 - 2 x 1 + 5 x 2 - x 3 + 4 x 4 = 1 3 x 1 - 7 x 2 + 4 x 3 - 4 x 4 = - 4 x 1 - 2 x 2 + x 3 - x 4 = - 2 - 2 x 1 + 5 x 2 - x 3 + 4 x 4 = 1 3 x 1 - 7 x 2 + 4 x 3 - 4 x 4 = - 4 -→ x 1 - 2 x 2 + x 3 - x 4 = - 2 x 2 + x 3 + 2 x 4 = - 3 x 3 + 1 2 x 4 = - 1 2 3

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Solution set: x 1 = - 13 2 - 3 2 a, x 2 = - 5 2 - 3 2 a, x 3 = 1 2 - 1 2 a, x 4 = a, a any real number . 4
Terms A matrix is a rectangular array of numbers. A matrix with m rows and n columns is an m × n matrix. Systems of linear equations: ma- trix representation: a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m 5

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Augmented matrix and matrix of coefficients: Augmented matrix: a 11 a 12 · · · a 1 n b 1 a 21 a 22 · · · a 2 n b 2 . . . . . . . . . . . . . . . a m 1 a m 2 · · · a mn b m Matrix of coefficients: a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . a m 1 a 32 · · · a mn 6
Elementary row operations: 1. Interchange row i and row j R i R j . 2. Multiply row i by a nonzero number k kR i R i . 3. Multiply row i by a number k and add the result to row j kR i + R j R j . 7

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Examples 1. Solve the system x + 2 y - 5 z = - 1 - 3 x - 9 y + 21 z = 0 x + 6 y - 11 z = 1 Augmented matrix: 1 2 - 5 - 1 - 3 - 9 21 0 1 6 - 11 1 Row reduce to: 1 2 - 5 - 1 0 1 - 2 1 0 0 1 - 1 8
Corresponding (equivalent) system of equations: x + 2 y - 5 z = - 1 y - 2 z = 1 z = - 1 Solution set: x = - 4 , y = - 1 , z = - 1 9

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2. Solve the system: 3 x - 4 y - z = 3 2 x - 3 y + z = 1 x - 2 y + 3 z = 2 Augmented matrix: 3 - 4 - 1 3 2 - 3 1 1 1 - 2 3 2 . Row reduce to: 1 - 2 3 2 0 1 - 5 - 3 0 0 0 1 10
Corresponding system of equations: x - 2 y + 3 z = 2 0 x + y - 5 z = - 3 0 x + 0 y + 0 z = 1 or x - 2 y + 3 z = 2 y - 5 z = - 3 0 z = 1 Solution set: no solution. 11

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3. Solve the system x + y - 3 z = 1 2 x + y - 4 z = 0 - 3 x + 2 y - z = 7 Augmented matrix: 1 1 - 3 1 2 1 - 4 0 - 3 2 - 1 7 Row reduce to: 1 1 - 3 1 0 1 - 2 2 0 0 0 0 . 12
Corresponding system of equations: x + y - 3 z = 1 0 x + y - 2 z = 2 0 x + 0 y + 0 z = 0 or x + y - 3 z = 1 y - 2 z = 2 0 z = 0 This system has infinitely many so- lutions given by: x = - 1 + a, y = 2 + 2 a, z = a, a any real number . 13

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Row echelon form: 1. Rows consisting entirely of ze- ros are at the bottom of the matrix. 2. The first nonzero entry in a nonzero row is a 1. This is called the leading 1. 3. If row i and row i + 1 are nonzero rows, then the leading 1 in row i +1 is to the right of the leading 1 in row i .
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