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Engr 231 Midterm 2 solutions

# Engr 231 Midterm 2 solutions - ENGR 231 Linear Engineering...

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ENGR 231 Fall 2011 Exam II Practice Problem Solutions 1 ENGR 231 Linear Engineering Fall 2011 Practice Problems Solutions for Exam II The exam will have questions like those on homework assignments 4, and 5. These problems dealt with material from sections 1.8, 1.9, 2.1, 2.2, 2.7. The material was also presented in lectures 7 through 10. Sections 1.8, 1.9. Problem 1. (a) Find the image under T of . T 1 1 1 ! " # # # \$ % & & & ' ( ) ) ) * + , , , = 1 - 5 - 7 - 3 7 5 ! " # \$ % & 1 1 1 ! " # # # \$ % & & & = - 11 9 ! " # \$ % & (b) Find a vector x such that T ( x ) = b . Is it unique? Answer : To find x , we solve the system. x = (3, 1). There is a free variable, so that the solution is not unique. (c) What are the domain and codomain of T ? Answer : The domain is R 3 , the codomain is R 2 . (d) What is the range of T ? Answer: Since there are two pivots, the range is the same as the codomain, namely R 2 . (e) Is T onto? Answer : The domain is the same as the codomain, so that T is onto. (f) Is T one-to-one? Answer : There is a free variable, so that there are infinite x values that map to one point. The transformation is not one-to-one. Problem 2. Answer True or False . Justify your answers. (a) Every matrix transformation is a linear transformation True . It follows from the distributive and scalar multiplication laws of matrix algebra. (b) The codomain of the transformation is R n , where n is the number of rows in A . True . A x is a vector in R n . By definition of the codomain, the codomain is R n .

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Engr 231 Midterm 2 solutions - ENGR 231 Linear Engineering...

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