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M21001-HW4 - s and verify by matrix multiplication that s...

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MATH 21001 - Spring 2010 - Homework 4 Hassan Allouba Problem 1 Each of the following is a function from < 2 into < 2 . Determine which are linear functions. (a) f x y = x 1 + y . (b) f x y = y x . (c) f x y = 0 xy . (d) f x y = x 2 y 2 . (e) f x y = x sin y . (f) f x y = x + y x - y . Problem 2 Determine the matrix associated with each of the following functions. That is, determine the a ij ’s such that f ( p ) = f x 1 x 2 = a 11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 . Where the functions are: (a) Reflect about the x -axis 1
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(b) Project onto the line y = x . Determine which of these transformations in < 2 are linear. Problem 3 By using matrix multiplication, represent the action of the linear function obtained by per- forming a reflection followed by a projection. Problem 4 Consider the following system of equations: 2 x 1 + x 2 + x 3 = 3 , 4 x 1 + 2 x 3 = 10 , 2 x 1 + 2 x 2 = - 2 . (a) Write the system as a matrix equation of the form Ax = b . 2
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(b) Write the solution of the system as a column s and verify by matrix multiplication
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Unformatted text preview: s and verify by matrix multiplication that s satisfies the equation Ax = b . (c) Write b as a linear combination of the columns in A . Problem 5 Let E = 1 0 0 0 1 0 3 0 1 and let A be an arbitrary 3 × 3 matrix. (a) Describe the rows of EA in terms of the rows of A . (b) Describe the columns of AE in terms of the columns of A . Problem 6 Let e j denote the j th unit column that contains a 1 in the j th position and zeros everywhere else. For a general matrix A n × n , describe the following products. (a) Ae j (b) e T i A (c) e T i Ae j Problem 7 Suppose that A and B are m × n matrices. If Ax = Bx holds for all n × 1 columns x , prove that A = B . Hint: What happens when x is a unit column? 3...
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