mela_exam1

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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Background image of page 2
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 ).
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 6 (12pts)
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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 ?...
View Full Document

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online