This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Partial Solution Set, Leon 4.1 First, recall that to show L is a linear transformation, one must show that either (1) L ( x + y ) = L ( x ) + L ( y ) and L ( x ) = L ( x ), or (2) L ( x + y ) = L ( x ) + L ( y ), or (3) L ( x + y ) = L ( x ) + L ( y ). Either one of the three would suffice. 4.1.1 For each of five transformations, we are to verify linearity and describe geometrically the effect of the transformation. Verification is straightforward for all; a sketch might make it easier to see whats going on geometrically. (a) L ( x ) = ( x 1 ,x 2 ) T . Verifying linearity is simple: we simply show that L ( x + y ) = ( ( x 1 + y 1 ) , ( x 2 + y 2 )) T = ( x 1 ,x 2 ) T + ( y 1 ,y 2 ) T = ( x 1 ,x 2 ) T + ( y 1 ,y 2 ) T = L ( x ) + L ( y ) . The geometric effect is reflection across the x 2axis. (c) Reflection across the identity line x 1 = x 2 . (e) Projection onto the x 2axis. 4.1.2 Let L be the linear transformation mapping R 2 into itself defined by L ( x ) = ( x 1 cos  x 2 sin ,x 1 sin + x 2 cos ) T . Express x 1 ,x 2 , and L ( x ) in terms of polar coordinates. Describe geometrically the effect of the linear transformation....
View
Full
Document
This note was uploaded on 07/17/2011 for the course MATH 311 taught by Professor Anshelvich during the Spring '08 term at Texas A&M.
 Spring '08
 Anshelvich
 Linear Algebra, Algebra

Click to edit the document details