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

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Unformatted text preview: 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....
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problemset7 - MAS 213: Linear Algebra II. Problem list for...

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