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

Info iconThis preview shows pages 1–2. 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: 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 ) = aδ ( v 1 , . . . , v i...
View Full Document

This note was uploaded on 12/05/2010 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

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