DOC PREVIEW
TAMU MATH 304 - Lect1-02web

This preview shows page 1-2-3-19-20-38-39-40 out of 40 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

MATH 304 Linear Algebra Lecture 2 Gaussian elimination continued Row echelon form Gauss Jordan reduction System of linear equations a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2n xn b2 am1 x1 am2 x2 amn xn bm Here x1 x2 xn are variables and aij bj are constants A solution of the system is a common solution of all equations in the system A system of linear equations can have one solution infinitely many solutions or no solution at all y x x y 2 2x 3y 6 x 0 y 2 y x 2x 3y 2 2x 3y 6 inconsistent system no solutions y x 4x 6y 12 2x 3y 6 2x 3y 6 Solving systems of linear equations Elimination method always works for systems of linear equations Algorithm 1 pick a variable solve one of the equations for it and eliminate it from the other equations 2 put aside the equation used in the elimination and return to step 1 x y 2 x y 2 2x y z 5 2 y 2 y z 5 After the elimination is completed the system is solved by back substitution y 1 x y 2 3 Gaussian elimination Gaussian elimination is a modification of the elimination method that allows only so called elementary operations Elementary operations for systems of linear equations 1 to multiply an equation by a nonzero scalar 2 to add an equation multiplied by a scalar to another equation 3 to interchange two equations Theorem i Applying elementary operations to a system of linear equations does not change the solution set of the system ii Any elementary operation can be undone by another elementary operation Operation 1 multiply the ith equation by r 6 0 a11 x1 a12 x2 a1n xn b1 ai1 x1 ai2 x2 ain xn bi a x a x a x b m1 1 m2 2 mn n m a11 x1 a12 x2 a1n xn b1 rai1 x1 rai2 x2 rain xn rbi a x a x a x b m1 1 m2 2 mn n m To undo the operation multiply the ith equation by r 1 Operation 2 add r times the ith equation to the jth equation ai1 x1 ai2 x2 ain xn bi a x a j1 1 j2 x2 ajn xn bj ai1 x1 ain xn bi a ra x ajn rain xn bj rbi j1 i1 1 To undo the operation add r times the ith equation to the jth equation Operation 3 interchange the ith and jth equations ai1 x1 ai2 x2 ain xn bi a x a x2 ajn xn bj j1 1 j2 aj1 x1 aj2 x2 ajn xn bj ai1 x1 ai2 x2 ain xn bi To undo the operation apply it once more Solution of a system of linear equations splits into two parts A elimination and B back substitution Both parts can be done by applying a finite number of elementary operations Example x y 2x y x y x 2 2 x y z 3 y z 1 z 6 2y z 4 3 y 2 x y 1 y z 1 z 2 3z 6 Another example x y 2z 1 y z 3 x 4y 3z 1 Add the 1st x y y 5y equation to the 3rd equation 2z 1 z 3 5z 2 Add 5 times the 2nd equation to the 3rd equation 1 x y 2z y z 3 0 13 System of linear equations x y 2z 1 y z 3 x 4y 3z 1 Solution no solution inconsistent system Yet another example x y 2z 1 y z 3 x 4y 3z 14 Add the 1st x y y 5y equation to the 3rd equation 2z 1 z 3 5z 15 Add 5 times the 2nd x y 2z y z 0 equation to the 3rd equation 1 3 0 Add 1 times the 2nd equation to the 1st equation z 2 x x z 2 y z 3 y z 3 0 0 Here z is a free variable x and y are leading variables x t 2 y t 3 for some t R It follows that z t System of linear equations x y 2z 1 y z 3 x 4y 3z 14 Solution x y z t 2 t 3 t t R In vector form x y z 2 3 0 t 1 1 1 Matrices Definition A matrix is a rectangular array of numbers 2 7 2 7 0 2 Examples 1 0 4 6 1 1 3 3 3 5 1 1 5 8 2 0 3 5 0 1 4 dimensions of rows of columns n by n square matrix n by 1 column vector 1 by n row vector System of linear equations a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2n xn b2 am1 x1 am2 x2 amn xn bm Coefficient matrix and column vector of the right hand sides a11 a12 a1n b1 a 21 a22 a2n b2 am1 am2 amn bm System of linear equations a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2n xn b2 am1 x1 am2 x2 amn xn bm Augmented a11 a12 a 21 a22 am1 am2 matrix a1n b1 a2n b2 amn bm Since the elementary operations preserve the standard form of linear equations we can trace the solution process by looking on the augmented matrix Elementary operations for systems of linear equations correspond to elementary row operations for augmented matrices 1 to multiply a row by a nonzero scalar 2 to add the ith row multiplied by some r R to the jth row 3 to interchange two rows Remark Rows are added and multiplied by scalars as vectors namely row vectors Elementary row operations Augmented a11 a 21 am1 matrix a12 a22 am2 a1n b1 v1 a2n b2 v2 vm amn bm where vi ai1 ai2 ain bi is a row vector Elementary row operations Operation 1 to multiply the ith row by r 6 0 v1 v1 vi r vi vm vm Elementary row operations Operation 2 to add the ith row multiplied by r to the jth row v1 v1 v v i i vj vj r vi vm vm Elementary row operations Operation 3 to interchange the ith row with the jth row v1 v1 v v j i vi vj vm vm Row echelon form Definition Leading entry of a matrix is the first nonzero entry in a row The goal of the Gaussian elimination is to convert the augmented matrix into row echelon form leading entries shift to the right as we go from the first row to the last one each leading entry is equal to 1 1 1 3 0 2 1 4 0 3 7 2 0 0 1 1 2 0 0 6 1 3 4 0 0 0 1 2 3 1 4 2 1 0 0 0 0 0 0 1 9 1 2 1 0 0 0 0 0 0 0 0 1 1 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 …


View Full Document

TAMU MATH 304 - Lect1-02web

Documents in this Course
quiz1

quiz1

2 pages

4-2

4-2

6 pages

5-6

5-6

7 pages

Lecture 9

Lecture 9

20 pages

lecture 8

lecture 8

17 pages

5-4

5-4

5 pages

Load more
Download Lect1-02web
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lect1-02web and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lect1-02web 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?