Applied Mathematics 1411A/B Chapter Notes - Chapter 6.1: Dot Product, Unit Vector, Identity Matrix

• School: Western University
• Department: Applied Mathematics
• Course: Applied Mathematics 1411A/B
• Professor: Bogdan Tudose

Document Summary

In this section we will use the most important properties of the dot product in rn and extend this notion of a dot product to all real vector spaces. Definition 1 - conditions of an inner product. {u, v} = {v, u} [ symmetry axiom] {u + v, w} = {u, w} + {v, w} [ additivity axiom] {v, v} 0 and {v, v} = 0 if and only if v = 0 [positivity axiom] A real vector space with an inner product is called a real inner product space. Because the inner product is just the dot product in rn to be: This is called the euclidean inner product (or the standard inner product) on rn to distinguish it from other possible inner products that might be defined on rn. Which means yes, there are other possible inner products. The rn associated with the euclidean inner product euclidean n-space. A vector of norm 1 is called a unit vector.