Exam2Solutions

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

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