problemset7 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #7. Tutorial on 26th October. This week’s topics: The change of co-ordinates matrix. Tutorial problems: Problem 1: (Problem 2.5.2, 2.5.3, various parts, from [FIS].) In the parts below you are given a vector space V and two bases for it, β and β . Find the change-of-coordinates matrix which takes β -coordinates to β -coordinates. (i) V = R 2 , β = { (1 , 0) , (0 , 1) } , and β = { ( a 1 , a 2 ) , ( b 1 , b 2 ) } . (ii) V = P 2 ( R ), β = { 2 x 2 - x, 3 x 2 + 1 , x 2 } and β = { 1 , x, x 2 } . (iii) V = P 2 ( R ), β = { x 2 - x, x 2 + 1 , x - 1 } , and β = { 5 x 2 - 2 x - 3 , - 2 x 2 + 5 x + 5 , 2 x 2 - x - 3 } . Problem 2: (Problem 2.5.4 from [FIS].) Let T : P 1 ( R ) P 1 ( R ) be the linear transformation which sends a polynomial to its derivative. Consider the following basis for this vector space: β = { 1 + x, 1 - x } . Use the “change-of-coordinates” theorem to calculate [ T ] β , starting from the matrix representation with respect to the standard ordered basis of P 1 ( R ). Then check your answer with a direct calculation of that matrix repre- sentation. 1
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Problem 3: The trace of an n × n -matrix is a familiar operation: tr( A ) = n i =1 A ii F . In this problem we want to extend the trace to abstract linear operators on finite-dimensional vector spaces.
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