Ma1bAnHw3

Ma1bAnHw3 - f-1 . 2. Let V be a nite dimensional vector...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b, Analytical; Homework Set 3 Due: 10am, Monday, January 28, 2008 You can collaborate on the problems as long as you write up all solutions in your own words and understand those solutions. Remember to justify all your answers. 1. Problem 26 on page 43 of Apostol. In part (b) prove that T and S are 1-1 correspondences. Recall from Lemma 1B that if f : V V is a function on a set V then f is a 1-1 correspondence iff f has an inverse function g . Further in that event by Lemma 1C, g is unique and denoted by
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f-1 . 2. Let V be a nite dimensional vector space of dimension n . For W V dene the codimension of W in V to be codim( W ) = dim( V )-dim( W ). Let W i , 1 i r , be subspaces of V and S = T r i =1 W i . Prove (1) codim( S ) r i =1 codim( W i ). (2) If r i =1 codim( W i ) < n then S 6 = 0. (Hint: Prove (1) by induction on r . In the case r = 2, use the Intersection/Sum Dimension Theorem from recitation section.) 3. Problem 28 on page 43 of Apostol. 1...
View Full Document

This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.

Ask a homework question - tutors are online