This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 3.5 Inner-Product Spaces 1 Chapter 3. Vector Spaces 3.5 Inner-Product Spaces Note. In this section, we generalize the idea of dot product to general vector spaces. We use this more general idea to define length and angle in arbitrary vector spaces. Note. Motivated by the properties of dot product on R n , we define the following: Definition 3.12. An inner product on a vector space V is a function that associates with each ordered pair of vectors ~v, ~w V a real number, written h ~v, ~w i , satisfying the following properties for all ~u,~v, ~w V and for all scalars r : P1. Symmetry: h ~v, ~w i = h ~w,~v i P2. Additivity: h ~u,~v + ~w i = h ~u,~v i + h ~u, ~w i , P3. Homogeneity: r h ~v, ~w i = h r~v, ~w i = h ~v, r ~w i , P4. Positivity: h ~v,~v i 0, and h ~v,~v i = 0 if and only if ~v = ~ . An inner-product space is a vector space V together with an inner product on V . 3.5 Inner-Product Spaces 2 Example. Dot product on R n is an example of an inner product: h ~v, ~w i = ~v...
View Full Document