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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #6 (due Thursday, October 20) All problems are from Leons book (8th edition). Section 3.5: 1b, 5a, 6a, 6b Section 3.6: 1a, 2a, 2b, 2c, 9, 16 MATH 304 Linear Algebra Lecture 14: Basis and coordinates. Change of basis. Linear transformations. Basis and dimension Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis . Theorem Any vector space V has a basis. If V has a finite basis, then all bases for V are finite and have the same number of elements (called the dimension of V ). Example. Vectors e 1 = (1 , , , . . . , , 0), e 2 = (0 , 1 , , . . . , , 0),. . . , e n = (0 , , , . . . , , 1) form a basis for R n (called standard ) since ( x 1 , x 2 , . . . , x n ) = x 1 e 1 + x 2 e 2 + + x n e n . Basis and coordinates If { v 1 , v 2 , . . . , v n } is a basis for a vector space V , then any vector v V has a unique representation v = x 1 v 1 + x 2 v 2 + + x n v n , where x i R . The coefficients x 1 , x 2 , . . . , x n are called the coordinates of v with respect to the ordered basis v 1 , v 2 , . . . , v n . The mapping vector v mapsto its coordinates ( x 1 , x 2 , . . . , x n ) is a onetoone correspondence between V and R n . This correspondence respects linear operations in V and in R n . Examples. Coordinates of a vector v = ( x 1 , x 2 , . . . , x n ) R n relative to the standard basis e 1 = (1 , , . . . , , 0), e 2 = (0 , 1 , . . . , , 0),. . . , e n = (0 , , . . . , , 1) are ( x 1 , x 2 , . . . , x n ). Coordinates of a matrix parenleftbigg a b c d parenrightbigg M 2 , 2 ( R ) relative to the basis parenleftbigg 1 0 0 0 parenrightbigg , parenleftbigg 0 0 1 0 parenrightbigg , parenleftbigg 0 1 0 0 parenrightbigg , parenleftbigg 0 0 0 1 parenrightbigg are ( a , c , b , d ). Coordinates of a polynomial p ( x ) = a + a 1 x + + a n 1 x n 1 P n relative to the basis 1 , x , x 2 , . . . , x n 1 are ( a , a 1 , . . . , a n 1 ). Vectors u 1 =(2 , 1) and u 2 =(3 , 1) form a basis for R 2 . Problem 1. Find coordinates of the vector v = (7 , 4) with respect to the basis u 1 , u 2 . The desired coordinates x , y satisfy v = x u 1 + y u 2 braceleftbigg 2 x + 3 y = 7 x + y = 4 braceleftbigg x = 5 y = 1 Problem 2. Find the vector w whose coordinates with respect to the basis u 1 , u 2 are (7 , 4)....
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This note was uploaded on 11/08/2011 for the course MATH 304 taught by Professor Hobbs during the Spring '08 term at Texas A&M.
 Spring '08
 HOBBS
 Linear Algebra, Algebra

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