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1bSet3

# 1bSet3 - f-1 2 Let V be a ﬁnite dimensional vector space...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b, Analytical; Homework Set 3 Due: 10am, Monday, January 25, 2010 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
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Unformatted text preview: f-1 . 2. Let V be a ﬁnite dimensional vector space of dimension n . For W ≤ V deﬁne 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...
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