{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapters_5_and_6-Gaussian Elim & Mesh Analysis

Chapters_5_and_6-Gaussian Elim & Mesh Analysis -...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Systems of Linear Equations: Gaussian Elimination For the sake of simplicity, we restrict ourselves to three or four unknowns. The basic ideas extend to arbitrary dimensions. Matrix Representation of a Linear System Matrices are helpful in rewriting a linear system in a very simple form. The algebraic properties of matrices may then be used to solve systems. First, consider the linear system 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
Set the matrices 2
Image of page 2
Using matrix multiplications, we can rewrite the linear system above as the matrix equation The matrix A is called the coefficient matrix . The matrix C is called the right-hand-side vector . X is the vector of unknowns. The augmented matrix is the matrix [ A | C ], where 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
ELEMENTARY OPERATIONS . It is clear that if we interchange two equations, the new system is still equivalent to the old one. If we multiply an equation with a nonzero number, we obtain a new system still equivalent to old one. 4
Image of page 4
And finally replacing one equation with the sum of two equations, we again obtain an equivalent system. These operations are called elementary operations Example. Consider the linear system The idea is to keep the first equation and work on the last two. In doing that, we will try to eliminate one of the 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
unknowns and solve for the other two. For example, if we keep the first and second equation, and subtract the first one from the last one, we get the equivalent system Next we keep the first and the last equation, and we subtract the first from the second. We get the equivalent system 6
Image of page 6
Now we focus on the second and the third equation. We repeat the same procedure. Try to eliminate one of the two unknowns ( y or z ). Indeed, we keep the first and second equation, and we add the second to the third after multiplying it by 3. We get This obviously implies z = -2. From the second equation, we get y = -2, and finally from the first equation we get x = 4. Therefore the linear system has one solution 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
Going from the last equation to the first while solving for the unknowns is called back substitution . Consider the augmented matrix corresponding to the original set of equations of the example.
Image of page 8
Image of page 9
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