Mulitvariate_Class_First_Part_Second_Week

Mulitvariate_Class_First_Part_Second_Week - Vector Space A...

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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 ! ! ! " "%
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# # # $ % & & & ( " & ( ) "" # * +’ ,* * !( & ( ! ! ! ! ! ! ! " " " ! ! " ! ! " ! ! " 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. ! ! " " ! ! ! " ) "&
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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 x’y 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 ! ! ! " ! ! ! " "+
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Geometric interpretation vector addition Geometrically, the addition of two vectors is simply starting one vector at the end of the other vector. "*
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DISTANCE Euclidean distance: As seen before, if we consider the point - . ! " x x P ! " in the plane, the straight-line distance, - 7 . P O d is: ! ! ! " - 7 . x x P O d ! " And ! ! ! ! ! " ### - 7 . p x x x x P O d ! ! ! ! " All points that lie a constant squared distance, such as , from the origin satisfy the equation !
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