Matrices n Vector Spaces

# Matrices n Vector Spaces - LINEAR ALGEBRA with Applications...

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LINEAR ALGEBRA with Applications BERNARD KOLMAN DAVID R. HILL

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MATRIX : An m x n matrix A is rectangular array of mn real numbers arranged in m horizontal rows and n vertical columns. Elementary Row Operations : 1. R i  R j 2. R i t R i 3. R i R i + t R j , t is non-zero real
Row Equivalent Matrix : An m x n matrix A is Row Equivalent to an m x n matrix B if B is obtained by applying finite sequence of elementary row operations to the matrix A.

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Results : Let A, B and C are m x n matrices. Then 1. A is row equivalent to A. 2. A is row equivalent to B, Implies, B is row equivalent to A 3. A is row equivalent to B, B is row equivalent to C, Implies, A is row equivalent to C
Example: 2 3 4 0 5 7 0 0 8 0 0 0

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Example: 0 3 4 5 6 0 0 7 6 5 0 0 0 8 9 0 0 0 0 0
Echelon Matrix : A matrix is Echelon Matrix if the number of zeros preceding the first non-zero entry ( called Leading element ) of a row increases row by row until only zero rows remains.

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Row Reduced Echelon Matrix : An Echelon matrix whose leading elements are 1.The only non-zero entries in their respective columns. 2.Each leading elements equals 1.
Ex: Row Reduced Echelon Matrix 0 1 3 0 0 4 0 0 0 0 1 0 3 0 0 0 0 0 1 2 0 0 0 0 0 0 0 1

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Pivot position: a position of a leading entry in an echelon form of the matrix. Pivot: a nonzero number that either is used in a pivot position to create 0’s or is changed into a leading 1, which in turn is used to create 0’s. Pivot column: a column that contains a pivot position.
PROBLEM : Find Row Reduced Echelon form of a Matrix 0 3 6 4 9 1 2 1 3 1 2 3 0 3 1 1 4 5 9 7 A - -     - - -   =   - - -   - -  

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Results : 1.Echelon form of a matrix may not be unique. 1.Row reduced echelon form of a matrix is unique 1. Every matrix is row equivalent to its echelon form. 1.Every matrix is row equivalent to its row reduced echelon form
Systems of Linear Equations A linear equation is an equation of the form Example Example of nonlinear equation is 1 1 2 2 n n a x a x a x b + + + = L ( 29 1 2 1 2 1 3 5 and 2 6 4 2 x x x x x x - = - + + = 1 2 1 2 3 rearranging 5 2 2 6 3 2 x x x x x ↓ ↓ - + - = =- 1 2 1 2 2 1 6 and 2 7 4 x x x x x x - = - =

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A linear system is a collection of one or more linear equations involving the same set of variables 1 2 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 1 i j n j n j n j n i j j i n n i i m a x a x a x a x b a x a x a x a x b a x a x a x a x b a x + + + + + + + = + + + + = + + + + = LL LL M M M LL M M M LL LL LL 1 2 2 m mj j mn n m a x a x a x b + + + + + = LL LL
= - - - - - - - - - - - - m i 2 1 n 2 1 mn 2 m 1 m in 2 i 1 i n 2 22 21 n 1 12 11 b b b b x x x a a a a a a a a a a a a M M M M M M M M M M M system of equation can be written in matrix form as A x =b,

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Matrices n Vector Spaces - LINEAR ALGEBRA with Applications...

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