m235notes6-1

# m235notes6-1 - Notes 6: Geometry of Linear Functions and...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 &amp;lt; v,u &amp;gt; = &amp;lt; v,av &amp;gt; +0 , so that a = &amp;lt;u,v&amp;gt; &amp;lt;v,v&amp;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....
View Full Document

## This note was uploaded on 11/22/2011 for the course MATH 235 taught by Professor Markman during the Fall '08 term at UMass (Amherst).

### Page1 / 3

m235notes6-1 - Notes 6: Geometry of Linear Functions and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online