Exam2Solutions

# Exam2Solutions - Newberger Math 247 Spring 03 Exam 2...

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Unformatted text preview: Newberger Math 247 Spring 03 Exam 2 Solutions 1. a. (7 points) Calculate the determinant of the following matrix. A = 6 0 0 5 2 0 0 1 7 2- 5 8 3 1 8 I expanded on the second row of A . det A = 2(- 1) 1+2 det 0 0 5 7 2- 5 3 1 8 = (- 2) · 5(- 1) 1+3 det • 7 2 3 1 ‚ = (- 10)(7 · 1- 2 · 3) =- 10 . b. (6 points) State the definition of one-to-one. A transformation T : R n → R m is called one-to-one if for every b ∈ R m , there exists at most one x ∈ R n such that T ( x ) = b . c. (6 points) State the definition of onto. A transformation T : R n → R m is called onto if for every b ∈ R m , there exists at least one x ∈ R n such that T ( x ) = b . 2. a. Let T be the transformation given by T ( x 1 ,x 2 ,x 3 ) = (- x 2 1 ,x 2 + x 3 ) . i. (2 points) What must a and b be so that T : R a → R b ? a = 3 and b = 2 . ii. (6 points) Is T linear? Explain why or why not. Your intuition should tell you this is not linear, since there is a square term in the first entry. However to show it is not linear, it is not enough to say that; you must show how this transformation fails to satisfy the definition of linear. First we check whether or not T ( ) = . If this were not true, then we would be done. Since it is truce we have to try something else. To show T is not linear, we must exhibit either two vectors, u and v , such that T ( u + v ) 6 = T ( u ) + T ( v ) , or a vector u and a scalar c such that T ( c u ) 6 = cT ( u ) . For example let u = (1...
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## This note was uploaded on 01/12/2010 for the course MATH 247 taught by Professor F,newberger during the Spring '03 term at Stanford.

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Exam2Solutions - Newberger Math 247 Spring 03 Exam 2...

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