Lecture_3_1 - Lecture Note 4 Sept 25 29 2006 Dr Jeff...

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Lecture Note 4: Sept 25 - 29, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu WONG 1
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MATRIX TRANSFORMATIONS Our aim is to study the notation R n for the set of all n -vectors with real entries. study the elements in R 2 and R 3 geometrically as directed line segments in a rectangular coordinate system. n -vector : A 1 × n or an n × 1 matrix is called an n -vector. When n is understood, we refer to n -vectors only as vectors. n -space : The set of all n -vectors is called n -space. For vectors whose entries are real numbers we denote n -space as R n . MATRIX TRANSFORMATIONS 2
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The vector x = x y in R 2 is represented by the directed line segment shown in Figure 1(a). The vector x = x y z in R 3 is represented by the directed line segment shown in Figure 1(b). MATRIX TRANSFORMATIONS 3
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O x y y - axis (x,y) x x - axis (a) x (x, y, z) x - z - y - axis axis axis O y x z (b) Figure 1: MATRIX TRANSFORMATIONS 4
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Example 1 Figure 2(a) shows geometric representations of the 2-vectors u 1 = 1 2 , u 2 = - 2 1 and u 3 = 0 1 in a two-dimensional rectangular coordinate system . Figure 2(b) shows geometric representations of the 3-vectors v 1 = 1 2 3 , v 2 = - 1 2 - 2 and v 3 = 0 0 1 in a three-dimensional rectangular coordinate system . MATRIX TRANSFORMATIONS 5
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u u x y - 2 O 1 1 2 u 1 2 3 (a) O y z x v v v 3 1 2 (b) Figure 2: MATRIX TRANSFORMATIONS 6
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Our aim here is: to give a short introduction from a geometrical point of view to certain functions mapping R n into R m . to study the so-called matrix transformation . Here we only consider the situation where m and n have values 2 or 3. MATRIX TRANSFORMATIONS 7
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Things to remember : A function is a rule that assigns to each element of one set exactly one element of another set. Definition: A transformation L of R n into R m , written L : R n R m , is a rule that assigns to each vector u in R n a unique vector v in R m . 1. R n is called the domain of L . 2. R m is the codomain . 3. We write L ( u ) = v ; (a) v is called the image of u under L . (b) The set of all images is called the range of L . The range may be used the whole of R n or only part of it. 4. The terms mapping and functions are also used for transformation. MATRIX TRANSFORMATIONS 8
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If A is an m × n matrix and u is an n -vector, then the matrix product A u is an m -vector. A function f mapping R n into R m is denoted by f : R n R m . A matrix transformation is a function f : R n R m defined by f ( u ) = A u . The vector f ( u ) in R m is called the image of u and the set of all images of the vectors in R n is called the range of f .
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