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Unformatted text preview: Lecture 2: Geometry vs Linear Algebra PointsVectors and DistanceNorm ShangHua Teng 2D Geometry: Points 2D Geometry: Cartesian Coordinates x y ( a,b ) 2D Linear Algebra: Vectors x y ( a,b ) 2D Geometry and Linear Algebra • Points • Cartesian Coordinates • Vectors 2D Geometry: Distance 2D Geometry: Distance How to express distance algebraically using coordinates??? Algebra: Vector Operations • Vector Addition • Scalar Multiplication + + = + = = 2 2 1 1 2 1 2 1 then and w v w v w v w w w v v v  = = 2 1 2 1 and 3 3 3 v v v v v v Geometry of Vector Operations • Vector Addition: v + w v w v + w Geometry of Vector Operations •v vv 2v Linear Combination Linear combination of v and w { c v + d w : c, d are real numbers} Geometry of Vector Operations • Vector Subtraction: v  w v w v + w v  w Norm: Distance to the Origin • Norm of a vector: 2 2 2 1   v v v + = 2 1 = v v v Distance of Between Two Points v w v  w ( 29 ( 29 2 2 2 2 1 1   ) , dist( w v w v w v w v + = = DotProduct (Inner Product) and Norm 2 2 1 1 w v w v w v + = • v v v • =   Angle Between Two Vectors v w θ Polar Coordinate v r φ ) sin , (cos ) sin , cos ( φ φ φ φ r r r v = = Dot Product: Angle and Length...
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 Spring '09
 BERA
 Linear Algebra, Algorithms, Vector Space, Linear combination, Vector Operations

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