mela_exam1

# mela_exam1 - Compute the determinant of A = 2-1 3 1-1 2-1 4...

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Math 107 (06) – Exam 1 – Spring 2009 Name: Problem 1 (10pts) . Complete the following deﬁnitions: (a) The vectors v 1 , . . . , v k are linearly independent iﬀ: (b) W is a subspace of the vector space V iﬀ: (c) The n × n matrix B is the inverse of the n × n matrix A iﬀ: (d) The vectors v 1 , v 2 and v 3 are linearly dependent iﬀ: (e) The set of vectors { v 1 , . . . , v k } form a basis iﬀ:

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Problem 2 (10pts) . Consider the vector space P 3 of all polynomials of degree at most 3. (a) Give an example of a basis for P 3 . (b) Is the subset W of all polynomials of degree exactly 2 a subspace of P 3 ? Problem 3 (12pts) . Find the inverse of the following matrix using the method of your choice (and check your answer): A = 1 - 1 2 - 3 .
Problem 4 (10pts) . Suppose that A and B are two n × n matrices with det( A ) = 3 and det( B ) = 7. Complete the following: (a) det( ABA ) = (b) det( A - 1 ) = (c) det( BAB - 1 ) = (d) det( A T B - 1 ) = (e) det(2 A ) = Problem 5 (12pts) . Consider the matrix A = 1 2 3 2 - 1 6 3 2 - 2 . (a) Find the LU -factorization of A (check your answer). (b) Use part (a) to ﬁnd det( A ).

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Problem 6 (12pts)

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Unformatted text preview: . Compute the determinant of A = 2-1 3 1-1 2-1 4 1-1 3 1 3 2-1 5 . Problem 7 (12pts) . Consider the linear transformation T • x y ‚ = • x + y x-y ‚ , from R 2 to R 2 ; and the vectors v 1 = • 1 2 ‚ and v 2 = •-1 1 ‚ . (a) Find the matrix A representing T in the standard basis { e 1 , e 2 } for both domain and range. (b) Find the matrix B representing T in the basis { v 1 , v 2 } for both domain and range. Problem 8 (12pts) . Is the vector b = 1 1 2 in the span of { 1 1 2 , 1 1 , 1 1 1 } ? Justify your answer. Problem 9 (10pts) . Let A be a 2 × 2 matrix. Suppose that A commutes with any 2 × 2 matrix (i.e. AM = MA for any 2 × 2 matrix M ). What can you say about A ?...
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mela_exam1 - Compute the determinant of A = 2-1 3 1-1 2-1 4...

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