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solution7

# solution7 - pp 105 Exercise 2 Let B = cfw_1 2 3 be the...

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Let B = { α 1 , α 2 , α 3 } be the basis for C 3 defined by α 1 = (1 , 0 , - 1) α 2 = (1 , 1 , 1) α 3 = (2 , 2 , 0) . Find the dual basis of B . The first element of the dual basis is the linear function α * 1 such that α * 1 ( α 1 ) = 1 , α * 1 ( α 2 ) = 0 and α * 1 ( α 3 ) = 0 . To describe such a function more explicitly we need to find its values on the standard basis vectors e 1 , e 2 and e 3 . To do this express e 1 , e 2 , e 3 through α 1 , α 2 , α 3 (refer to the solution of Exercise 1 pp. 54-55 from Homework 6). For each i = 1 , 2 , 3 you will find the numbers a i , b i , c i such that e 1 = a i α 1 + b i α 2 + c i α 3 (i.e. the coordinates of e i relative to the basis α 1 , α 2 , α 3 ). Then by linearity of α * 1 we get that α * 1 ( e i ) = a i . Then α * 2 ( e i ) = b i , and α * 3 ( e i ) = c i . This is the answer. It can also be reformulated as follows. If P is the transition matrix from the standard basis e 1 , e 2 , e 3 to α 1 , α 2 , α 3 , i.e. ( α 1 , α 2 , α 3 ) = ( e 1 , e 2 , e 3 ) P , then ( P - 1 ) t is the transition matrix from the dual basis e * 1 , e * 2 , e * 3 to the dual basis α * 1 , α * 2 , a * 3 , i.e. ( α * 1 , α * 2 , a * 3 ) = ( e * 1 , e * 2 , e * 3 )( P - 1 ) t . Note that this problem is basically the change of coordinates problem: e.g. the value of α * 1 on the vector v C 3 is the first coordinate of v relative to the basis α 1 , α 2 , α 3 .

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solution7 - pp 105 Exercise 2 Let B = cfw_1 2 3 be the...

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