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Lect2-07-web

# Lect2-07-web - MATH 304 Fall 2011 Linear Algebra Homework...

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MATH 304, Fall 2011 Linear Algebra

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Homework assignment #6 (due Thursday, October 20) All problems are from Leon’s 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.

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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 , 0 , . . ., 0 , 0), e 2 = (0 , 1 , 0 , . . ., 0 , 0),. . . , e n = (0 , 0 , 0 , . . ., 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 one-to-one correspondence between V and R n . This correspondence respects linear operations in V and in R n .

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Examples. Coordinates of a vector v = ( x 1 , x 2 , . . ., x n ) R n relative to the standard basis e 1 = (1 , 0 , . . ., 0 , 0), e 2 = (0 , 1 , . . ., 0 , 0),. . . , e n = (0 , 0 , . . ., 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 0 + a 1 x + · · · + a n 1 x n 1 ∈ P n relative to the basis 1 , x , x 2 , . . ., x n 1 are ( a 0 , 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). w = 7 u 1 + 4 u 2 = 7(2 , 1) + 4(3 , 1) = (26 , 11)

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Change of coordinates Given a vector v R 2 , let ( x , y ) be its standard coordinates, i.e., coordinates with respect to the standard basis e 1 = (1 , 0), e 2 = (0 , 1), and let ( x , y ) be its coordinates with respect to the basis u 1 = (3 , 1), u 2 = (2 , 1). Problem. Find a relation between ( x , y ) and ( x , y ).
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Lect2-07-web - MATH 304 Fall 2011 Linear Algebra Homework...

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