Lecture7 - University of Toronto at Scarborough Department...

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 7 Sophie Chrysostomou 1.4 Solving Systems of Linear Equations The Connection Between Linear Systems of Equations and Matrices Suppose that a11 a12 · · · a1n a21 a22 · · · a2n A= . . . . . . . . . . . . am1 am2 · · · amn , x= x1 x2 . . . ,b = b1 b2 . . . bm xn what is the meaning of Ax = b? definition 0.1. Suppose we have the linear system a11 x1 + a12 x2 + ... + a1n xn a21 x1 + a22 x2 + ... + a2n xn . . . . . . . . . . . . = = . . . b1 b2 . . . am1 x1 + am2 x2 + ... + amn xn = bm This is equivalent to Ax = b a11 a12 · · · a21 a22 · · · A= . . . . . . . . . am1 am2 · · · where a1 n a2 n . , . . amn 1 x= x1 x2 . . . xn ,b = b1 b2 . . . bm A is called the coefficient matrix a11 a21 A= . . . am1 of the system and · · · a1 n · · · a2 n . . . . . . · · · amn a12 a22 . . . am2 b1 b2 . . . bm is called the augmented or partitioned matrix of the system. We may solve the linear system using the operations: 1. Switch two equations. 2. Multiply an equation by a nonzero constant. 3. Replace an equation by that equation plus a multiple of another equation. OR we can use the augmented matrix and use the following operations : Elementary Row Operations a) (Row Interchange) Interchange the ith and j th row (Ri ←→ Rj ) vectors of the matrix. b) (Row scaling) Multiply the ith row of the matrix with a (Ri → aRi ) nonzero scalar a c) (Row addition) Add to the ith row of the matrix s times (Ri → R1 + sRj ) the j th row. definition 0.2. If a matrix B can be obtained from A by performing a sequence of elementary operations, then A and B are called row equivalent. theorem 0.3. If [A|b] and [H |a] are row equivalent matrices then Ax = b and H x = a have the same solution set. 2 example 0.4. Find the solution to the system 2x1 + 3x2 = 9 x1 − 2x2 = 1 definition 0.5. Row Echelon Form, Pivot. A matrix is in a row echelon form (REF) if: a) All rows containing only zeros are below rows with nonzero entries. b) The first nonzero entry in any row appears in a column to the right of the first nonzero entry in any row above it. If a matrix is in a row echelon form, then the first nonzero entry in a row is called the pivot of that row. example 0.6. 3 definition 0.7. Reduced Row Echelon Form, Pivot. A matrix is in a reduced row echelon form (RREF) if: a) The matrix is in a row echelon form b) Every pivot equals 1. c) Each pivot is the only nonzero entry in its column. example 0.8. example 0.9. Find the solution to the system 2x1 − x2 − 3x3 = 4 x1 + 2x2 + 2x3 = 6 3x1 + x2 + x3 = 8 4 example 0.10. Find the solution to the system x1 + 2x2 + 2x3 = 6 2x1 − x2 − 3x3 = 4 3x1 + x2 − x3 = 8 5 example 0.11. Find the solution to the system x1 + 2x2 + 2x3 = 6 2x1 − x2 − 3x3 = 4 3x1 + x2 − x3 = 10 definition 0.12. Consistent Linear System. A linear system having one or more solutions is called a consistent system. If it has no solutions, then it is called an inconsistent system. definition 0.13. The Gauss reduction with back substitution method refers to the method of solving the system Ax = b by reducing [A|b] to a row echelon form and then using back substitution. The Guass-Jordan method refers to the method of solving the system Ax = b by reducing [A|b] to a reduced row echelon form. 6 ...
View Full Document

This note was uploaded on 05/22/2010 for the course MATH MATA23 taught by Professor Sophie during the Spring '10 term at University of Toronto- Toronto.

Ask a homework question - tutors are online