Inner Products Space

Inner Products Space

1. Vector Length & Unit Vectors

   • The length, or norm, of a vector v in R^n is given by
     • Length of a scalar multiple
     • Unit vector in the direction of v-bar
   • The distance of two vectors
     • Satisfy 4 properties
   • Dot product in R^n
     • Properties of dot product
   • Orthogonal vectors
     • Cauchy-Schwarz inequality
   • Angle between 2 vectors
     • Phytagoran theorem
     • Triangle inequality

2. Inner Products Space

   • Satisfy 4 axioms
     • • Symmetry axiom
     • • Additivity axiom
     • • Homogeneity axiom
     • • Positivity axiom
   • If V is an inner product space, then
     • norm
     • If u-bar and v-bar are vectors of R^n with the Euclidean IP, then
     • distance
   • Weighted Euclidean inner product with weights

3. Orthogonal Basis

   • Orthogonal
     • An orthogonal set of nonzero vectors in R^n is linearly independent
   • An orthogonal (orthonormal) basis for a vector space is a basis that is orthogonal (orthonormal) set
     • Let S={vectors u}, an orthonormal basis for R^n and v any vector in R&n
     • Gram-Schmidt Theorem
       • S3 :
       • S1 : Let
       • S4 :
       • S2 :
   • Orthonormal

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