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Week7 - Supplement to Week 5 March 6 2007 0.1 Determinants...

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Supplement to Week 5 March 6, 2007 0.1 Determinants Let F be a field, let n be a positive integer, and let V := F n . The geo- metric idea is the following. Given a sequence ( v 1 , . . . v n ) of n vectors in V , δ ( v 1 , . . . v n ) is supposed to be some element of F which corresponds to some notion of the “oriented volume” determined by this list of vectors. The order of the vectors may matter, because of the “orientation.” Although it is not clear a priori what this should mean, it turns out (amazingly!) that three simple properties that seem intuitively natural for such a notion to satisfy determine it uniquely. Advice: write these out when n = 2, and draw pictures of parallelograms to see what they mean geometrically. Theorem 1: With the notation above, there is a unique function δ : V n F satisfying the following properties: 1. For any ( v 1 , v 2 , . . . , v n ) V n , any i , and any a F , δ ( v 1 , . . . , av i , . . . , v n ) = ( v 1 , . . . , v i , . . . , v n ) . 2. If i = j , and v i = v i + v j , then δ ( v 1 , . . . , v i . . . , v n ) = δ ( v 1 , . . . , v i , . . . , v n ) .

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