c3s5 - 3.5 Inner-Product Spaces 1 Chapter 3. Vector Spaces...

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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...
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c3s5 - 3.5 Inner-Product Spaces 1 Chapter 3. Vector Spaces...

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