Lesson 9b -Homogenous Systems ^ Particular Solution

Lesson 9b -Homogenous Systems ^ Particular Solution -...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
23/09/2013 1 Homogenous System A Homogenous System is a linear system Ax=b, where b = 0 (the vector of all zeros). A homogenous system is always consistent! (It may be one solution, or infinitely many solutions, but never no solution). Why? Ax=0 will always have the solution x = 0 (A matrix multiplied by the zero vector will always produce the zero vector). Solution Sets for Homogenous Systems Theorem: Any homogenous system with k free variables will have the following solution set: } ,..., , | ... { 2 1 2 2 1 1 R t t t v t v t v t S k k k Where our t i are the free variables, and our v i are non zero vectors. Why? Well any free variable (x j ) we must substitute x j = t i (depending on the free variable location, we may have to skip some basic variables before getting to the next free variable). This means for each free variable, we introduce a new parameter, and the vector (v i ) will (at minimum) have the jth entry as a 1.
Background image of page 1

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

View Full Document Right Arrow Icon
23/09/2013 2 Example If we row reduce a matrix and it looks like the following: 0 0 1 0 2 1 0 2 0 1 Here we see that x 3 and x 4 are free. Which means we will assign them each a different free variable. In this case let us call them x 3 =t 1 and x 4 =t 2 . Right away we can see that the number of free variables will match the number of parameters we have. If we continue to solve for the basic variables we get: x 1 = 2t 1 and x 2 = 2t 1 t 2 . This gives us the solution set as follows: 0 0 0 0 0
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

Lesson 9b -Homogenous Systems ^ Particular Solution -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online