Understanding Inner Products in Vector Spaces
Learn about inner products in vector spaces, including definitions, properties, and examples. Explore how inner products are used to define a function that maps vectors to numbers in a linear fashion. Discover the fundamental characteristics of inner product spaces and their basic properties.
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Presentation Transcript
INNER PRODUCT A.G.Farhan . . .
DEFINITION1: Definition1 An inner product on a vector space V on F is a function: , that takes each two vectors u and v of V to a number and has the following properties: 1) Positivity: 2) Definiteness 3) Additivity in first slot:
4)homogeneity in first slot: 5) Conjugate symmetry: : such that geometrically
If EXAMPLES 1. then is inner product on a vector space on 2. If are positive numbers, then an inner product can be defined on by:
DEFINITION2 An inner product space is a vector space V along with an inner product on V. ; BASIC PROPERTIES OF AN INNER PRODUCT (a) For each fixed u V the function that takes v to is a linear map from V to F. (b) (c) (d) (e)
