Math 115A: Linear Algebra Homework 6: Final version: Due Friday November 4 Homework from text:4 4, 6, 7, 17, 2.5 2d, 4, 5, 7.1 3, 8, 17, 20. Double...
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i need details of every step of double star problem( 11, 12, 13). 

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Math 115A: Linear Algebra Homework 6: Final version: Due Friday November 4 Homework from text: § 2.4 4, 6, 7, 17, § 2.5 2d, 4, 5, 7. § 5.1 3, 8, 17, 20. Double star 11: Let P n be all real polynomials of degree less than or equal to n . Let c 0 ,c 1 ,...c n be distinct numbers. Let T : P n R n +1 be defined by T ( p ( x )) = ( p ( c 0 ) ,p ( c 1 ) ,...,p ( c n )) . Show T is an isomorphism. You may use that a polynomial p ( x ) P n has at most n roots unless p ( x ) = 0. The fact that T is onto is called Lagrange interpolation, famous in applied math. Double star 12: Suppose V has dimension 2 and T : V V is linear. Suppose T 2 = T . Suppose T is not 0 and that T is not an isomorphism. Show V has a basis { α 1 2 } = B so that [ T ] B = 1 0 0 0 ! Hint: Show there are α 1 R ( T ), α 2 N ( T ) which form a basis. Double star 13: Suppose dim V = 3 and T 3 = 0, but T 2 6 = 0. Show that we can find a basis B so that [ T ] B = 0 0 0 1 0 0 0 1 0 Hint: If T 2 ( α ) 6 = 0, consider B = { α,T ( α ) ,T 2 ( α ) } . You have to show B is a basis. Problem: In this problem, the scalar field is C . Let A = cos( θ ) - sin( θ ) sin( θ ) cos( θ ) !
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Find a matrix Q so that Q - 1 AQ is diagonal. What is Q - 1 AQ ? Challenge problems: Text § 2.5: 13. Challenge 6.1 A variant of double star 12: Suppose V has dimension 2 and T : V V is linear. Suppose T 2 = T . Suppose T is not 0 and that T is not the identity. Show V has a basis { α 1 2 } = B so that [ T ] B = 1 0 0 0 ! Challenge 6.2 Suppose dim V = 3 and T 2 = 0, but T 6 = 0. Show there is a basis B so that [ T ] B = 0 0 0 1 0 0 0 0 0
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