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Unformatted text preview: Notes 6: Geometry of Linear Functions and Multiplication of Matrices Let v = (1 , 2) R 2 . Define the function proj v : R 2- R 2 to be the orthogonal projection onto the line through v . What is the matrix representing this function? By definition proj v ( u ) = av where u = av + x,a R 2 and x is orthogonal to v . This means that &lt; v,u &gt; = &lt; v,av &gt; +0 , so that a = &lt;u,v&gt; &lt;v,v&gt; . Writing u = ( u 1 ,u 2 ), we see that a = (1 / 5) u 1 + (2 / 5) u 2 , and hence proj v ( u 1 ,u 2 ) = av = ( u 1 / 5 + (2 / 5) u 2 , (2 / 5) u 1 + (4 / 5) u 2 ) . From here there are two ways of reading off the matrix of proj v . Note that the function ( x,y ) 7 ( ax + by,cx + dy ) is given by the matrix a b c d , so that the matrix of proj v is 1 / 5 2 / 5 2 / 5 4 / 5 . another way to proceed is to calculate proj v ( e 1 ) ,proj v ( e 2 ) . These give the columns of our sought after matrix. Products of Matrices Let f : S- T,g : T- U be two functions. The compostion of f and g is denoted by g f and is a function with domain S and target U . It acts on an element....
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