Mulitvariate_Class_First_Part_Second_Week

Mulitvariate_Class_First_Part_Second_Week - Vector Space A...

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Unformatted text preview: Vector Space A vector consisting of m elements may be regarded geometrically as a point in m- dimensional space. A vector consisting of p elements can also be regarded geometrically as a line from the origin to a point in p-dimensional space. Length of a vector The length of a vector of m elements emanating from the origin is given be the Pythagorean formula: Length of ! ! ! ! " ####### m x x x x L ! ! ! " "% # # # $ % & & & ( " & ( ) "" # * + ,* * !( & ( ! ! ! ! ! ! ! " " " ! ! " ! ! " ! ! " x x x L x Angle between two vectors The angle ) between two vectors x and y , both having m entries is defined from y x L L m m y x y x y x- ####### .- /01. ! ! " " ! ! ! " ) "& So # # # # $ % & & & & ( * * " ! ! ( " ) # # # # $ % & & & & ( * " " 2 , 3 + # ( " x L "2 # ( " y L + " * " * ! ! * ! * " ! ! ! " "( &2% #- "2 # (-. + # ( . 4 "- ! .- 2 . !- . ( ,- " 5.- .- /01. , , ! ! " " ) ) y x L L y x y x y x y x The angle in between two vectors using the xy notation y y x x y x 6 6 6- /01. " ) When the angle between two vectors x and y is + + " !&2 *2 or ) then x and y are perpendicular. Geometric Interpretation scalar multiplication If a m-dimensional vector x of length is multiplied by a scalar, then the length of the resulting vector is ! ! ! ! " ! ! ! ! ! ! " ! ####### ####### m m x x x x c x c x c x c cL ! ! ! " ! ! ! " "+ Geometric interpretation vector addition Geometrically, the addition of two vectors is simply starting one vector at the end of the other vector. "* DISTANCE Euclidean distance: As seen before, if we consider the point - ....
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Mulitvariate_Class_First_Part_Second_Week - Vector Space A...

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