section34 - Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3...

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Unformatted text preview: Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3 Lecture Notes Fall 2007 Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 Chapter 3 3.4: Matrix Transformations of R 2 Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.4: Matrix Transformations of R 2 Linear Transformations A linear transformation T : R 2- R 2 is a mapping that takes vectors in R 2 to vectors in R 2 such that T ( ~ u + ~ v ) = T ( ~ u ) + T ( ~ v ) and T ( k ~ v ) = kT ( ~ v ) , for all vectors ~ u and ~ v , and all scalars k . This definition is equivalent to saying T ( a ~ u + b ~ v ) = aT ( ~ u ) + bT ( ~ v ) for all vectors ~ u and ~ v , and all scalars, a and b . We say, T preserves linear combinations. Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.4: Matrix Transformations of R 2 Example 1 Let T ( ~ v ) = 2 ~ v . Then T ( ~ u + ~ v ) = 2( ~ u + ~ v ) = 2 ~ u + 2 ~ v = T ( ~ u ) + T ( ~ v ) and T ( k ~ v ) = 2( k ~ v ) = k (2 ~ v ) = kT ( ~ v ) . So T is a linear transformation. Indeed, any scalar map, T ( ~ v ) = a ~ v , for any scalar a , is a linear transformation, as you can check. Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.4: Matrix Transformations of R 2 Example 2 A less obvious example is the projection map. Let ~ d be any non-zero vector in R 2 , and define T : R 2- R 2 by T ( ~ v ) = proj ~ d ( ~ v ) . Then Then T ( ~ u + ~ v ) = ( ~ u + ~ v ) ~ d ~ d ~ d ~ d = ~ u ~ d + ~ v ~ d ~ d ~ d ~ d = ~ u ~ d ~ d ~ d ~ d + ~ v ~ d ~ d ~ d ~ d = T ( ~ u ) + T ( ~ v ) and T ( k ~ v ) = ( k ~ v ) ~ d ~ d ~ d ~ d = k ~ v ~ d ~ d ~ d ~ d = kT ( ~ v ) . Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.4: Matrix Transformations of R 2 Example 3 Let T x y = 2 x + 3 y- x + y . Then T is a linear transformation; the easy way to see this is to rewrite T in terms of matrix multiplication, and then use properties of matrices. T x y = 2 x + 3 y- x + y = 2 3- 1 1 x y . Let A = 2 3- 1 1 ; ~ v = x y . Then T ( ~ v ) = A ~ v and T ( ~ u + ~ v ) = A ( ~ u + ~ v ) = A ~ u + A ~ v = T ( ~ u ) + T ( ~ v ); also T ( k ~ v ) = A ( k ~ v ) = kA ~ v = kT ( ~ v ) . So T is a linear transformation, known as a matrix transformation. Chapter 3 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 3 3.4: Matrix Transformations of R 2 Matrix Transformations In fact every linear transformation T : R 2- R 2 is a matrix transformation. To see this, let ~ i = 1 ; ~ j = 1 and ~ v = x y . Then ~ v = x ~ i + y ~ j and T ( ~ v ) = T ( x ~ i + y ~ j ) = xT ~ i + yT ~ j = A ~ v , for A = h T ~ i | T ~ j i . A is called the standard matrix of T ; and from now on we will consider all linear transformations as matrix transformations....
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This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto- Toronto.

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section34 - Chapter 3 MAT188H1F Lec03 Burbulla Chapter 3...

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