math355-fall05-hw7

math355-fall05-hw7 - v 1 v 2-v 3 and v 1 2 v 2 v 3 are...

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MATH 355 Section 001: Linear Algebra Fall 2005 Homework 7, due Friday 10/21 1. p. 205, problem 6. 2. Let B := { v 1 ; v 2 ; v 3 } be an ordered basis for a vector space V . What is c B (0) , c B ( - v 2 ) , c B ( v 1 + v 3 )? 3. Let B 1 := 0 1 0 ; 1 0 0 ; 0 0 1 , B 2 := 0 1 0 ; 0 0 1 ; 1 0 0 . (a) What is c B 1 ( v ) and c B 2 ( v ) for v = ( v 1 v 2 v 3 ) t ?. (b) What is the base change matrix from B 2 to B 1 , i.e. what is the matrix M such that c B 1 ( v ) = Mc B 2 ( v ) for all v ? 4. Let B = { v 1 ; v 2 ; v 3 } be an ordered basis. Let ˜ B := { v 1 + v 2 ; v 3 ; v 2 } . What is the base change matrix from B to ˜ B , i.e. what is the matrix M such that c B ( v ) = Mc ˜ B ( v ) for all v ? 5. p. 205, problems 15 (a), (b) and 16 (a), (b). 6. Let B := { v 1 ; v 2 ; v 3 } be an ordered basis for a vector space V . (a) Use the principle of isomorphism to show that
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Unformatted text preview: v 1 + v 2-v 3 and v 1 + 2 v 2 + v 3 are linearly independent. (Hint: ﬁrst ﬁnd c B ( v 1 + v 2-v 3 ) and c B ( v 1 + 2 v 2 + v 3 )). (b) Using the principle of isomorphism determine whether v 1 + v 2-v 3 , 2 v 1 + 3 v 2 and v 1 + 2 v 2 + v 3 are linearly independent. 7. Are the polynomials 1 + t + t 3 , 1-2 t + 3 t 3 , 2-t + t 3 linearly independent in the vector space V := { polynomials of degree less or equal than 3 } . Hint: Pick an ordered basis for V and then apply the principle of isomorphism....
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