linear lectures - Matrix and Numerical Methods in Systems...

Info icon This preview shows pages 1–22. Sign up to view the full content.

View Full Document Right Arrow Icon
Matrix and Numerical Methods in Systems Engineering
Image of page 1

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

View Full Document Right Arrow Icon
Linear Algebra
Image of page 2
Systems of Linear Equations I Linear equation: ax + by = c I variables: x , y I constants: a , b , c I System of linear equations 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 I n variables I m equations I nm constants
Image of page 3

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

View Full Document Right Arrow Icon
Example 1 2 x - y = 0 - x + 2 y = 3
Image of page 4
Example 1 2 x - y = 0 - x + 2 y = 3 I Matrix form 2 - 1 - 1 2 x y = 0 3 or just A x = b
Image of page 5

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

View Full Document Right Arrow Icon
Example 1 2 x - y = 0 - x + 2 y = 3 I Matrix form 2 - 1 - 1 2 x y = 0 3 or just A x = b I Three interpretations I row I column I matrix
Image of page 6
Example 1 I Row I 2 x - y = 0 and - x + 2 y = 3 I solution: x = 1, y = 2 I number of solutions
Image of page 7

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

View Full Document Right Arrow Icon
Example 1 I Row I 2 x - y = 0 and - x + 2 y = 3 I solution: x = 1, y = 2 I number of solutions I Column I linear combination x 2 - 1 + y - 1 2 = 0 3 I solution: x = 1, y = 2 I number of solutions
Image of page 8
Example 1 I Row I 2 x - y = 0 and - x + 2 y = 3 I solution: x = 1, y = 2 I number of solutions I Column I linear combination x 2 - 1 + y - 1 2 = 0 3 I solution: x = 1, y = 2 I number of solutions I Matrix 2 - 1 - 1 2 2 1 = 2 2 - 1 + 1 - 1 2 = 3 0
Image of page 9

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

View Full Document Right Arrow Icon
Example 2 2 x - y =0 - x +2 y - z = - 1 - 3 y +4 z =4
Image of page 10
Example 2 2 x - y =0 - x +2 y - z = - 1 - 3 y +4 z =4 I Row: planes instead of lines
Image of page 11

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

View Full Document Right Arrow Icon
Example 2 2 x - y =0 - x +2 y - z = - 1 - 3 y +4 z =4 I Row: planes instead of lines I Column x 2 - 1 0 + y - 1 2 - 3 + z 0 - 1 4 = 0 - 1 4 I Solution: x = y = 0, z = 1
Image of page 12
Example 2 2 x - y =0 - x +2 y - z = - 1 - 3 y +4 z =4 I Row: planes instead of lines I Column x 2 - 1 0 + y - 1 2 - 3 + z 0 - 1 4 = 0 - 1 4 I Solution: x = y = 0, z = 1 I A x = b – Does a solution always exist?
Image of page 13

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

View Full Document Right Arrow Icon
Elimination I Natural I Two stages I forward elimination I backward substitution
Image of page 14
Elimination I Natural I Two stages I forward elimination I backward substitution I Example x +2 y + z =2 3 x +8 y + z =12 4 y + z =2
Image of page 15

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

View Full Document Right Arrow Icon
Elimination I Natural I Two stages I forward elimination I backward substitution I Example x +2 y + z =2 3 x +8 y + z =12 4 y + z =2 I Forward step: pivots A = 1 2 1 3 8 1 0 4 1 1 2 1 0 2 - 2 0 4 1 1 2 1 0 2 - 2 0 0 5 = U I U – upper triangular matrix
Image of page 16
Elimination I What can go wrong I pivots 1 & 2 equal to 0: interchange rows I pivot 3 equals to 0: no way out
Image of page 17

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

View Full Document Right Arrow Icon
Elimination I What can go wrong I pivots 1 & 2 equal to 0: interchange rows I pivot 3 equals to 0: no way out I Augmented matrix 1 2 1 2 3 8 1 12 0 4 1 2 1 2 1 2 0 2 - 2 6 0 4 1 2 1 2 1 2 0 2 - 2 6 0 0 5 - 10 = [ U c ] x +2 y + z =2 2 y - 2 z =6 5 z = - 10
Image of page 18
Elimination I What can go wrong I pivots 1 & 2 equal to 0: interchange rows I pivot 3 equals to 0: no way out I Augmented matrix 1 2 1 2 3 8 1 12 0 4 1 2 1 2 1 2 0 2 - 2 6 0 4 1 2 1 2 1 2 0 2 - 2 6 0 0 5 - 10 = [ U c ] x +2 y + z =2 2 y - 2 z =6 5 z = - 10 I Backward substitution: z = - 2, y = 1, x = 2
Image of page 19

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

View Full Document Right Arrow Icon
Matrix Multiplication I matrix-by-column A b 1 b 2 b 3 = b 1 a 11 a 21 a 31 + b 2 a 12 a 22 a 32 + b 3 a 13 a 23 a 33
Image of page 20
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