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5 Cards in this Set
- Front
- Back
The standard inner product, dot product, of two vectors x and y in R^n
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<x∣y>=(x^T)(y)=x₁y₁+x₂y₂+...xₐyₐ
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the inner product in R has four properties
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1)Linearity 1st factor <c₁x+c₂y∣z>=c₁<x∣z>+c₂<y∣z>
2)Linearity 2nd factor <x∣c₁y+c₂z>=c₁<x∣y>+c₂<x∣z> 3)Symmetry: <x∣y>=<y∣x> 4)Positivity: If x≠0, then <x∣x> > 0. |
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define the length, or norm of a vector
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∣x∣=√<x∣x>
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Schwarz inequality
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Let V be a real inner product space and let x and y be vectors in V. Then <x∣y>≤∣x∣∣y∣, with equality only if x and y are linearly dependent.
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Schwarz inequality proof
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-If x and y are linearly dependent, then one is a multiple of the other, and the equality follows from linearity.
-Suppose x and y linearly independent. Define family of vectors v(t)=x+ty where t ranges from -∞ to ∞ and v(t)≠0 0<∣v(t)∣= <v(t)∣v(t)>= <x+ty∣x+ty>= t^2<y∣y> + 2t<x∣y> + <x∣x> Plug-in t=-<x∣y>/<y∣y> 0 < (<x∣y>^2/<y∣y>)-(2<x∣y>^2/<y∣y>)+<x∣x> =(<x∣x><y∣y>-<x∣y>^2)/<y∣y> Then ∣x∣∣y∣=√<x∣x><y∣y> > <x∣y> |