{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 11

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 13

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern