Lecture_13 - Lecture Note 13 Dr. Jeff Chak-Fu WONG...

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Unformatted text preview: Lecture Note 13 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1 • Matrix Transformations • Linear Transformations Produced by Jeff Chak-Fu WONG 2 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 3 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 4 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 5 Example 1 Figure 2(a) shows geometric representations of the 2-vectors u 1 = 1 2 , u 2 = - 2 1 and u 3 = 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 = 1 in a three-dimensional rectangular coordinate system . MATRIX TRANSFORMATIONS 6 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 7 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 8 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 . 4. The terms mapping and functions are also used for transformation. MATRIX TRANSFORMATIONS 9 • 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 ....
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Lecture_13 - Lecture Note 13 Dr. Jeff Chak-Fu WONG...

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