Lecture_14 - Lecture Note 14 Dr Jeff Chak-Fu WONG...

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Unformatted text preview: Lecture Note 14 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1 L INEAR T RANSFORMATIONS AND M ATRICES In Lecture note 13, we gave the definition, basic properties, and some examples of linear transformations mapping R n into R m . In this note we consider linear transformations mapping a vector space V into a vector space W . LINEAR TRANSFORMATIONS AND MATRICES 2 1. Definition and Examples 2. The Kernel and Range of a Linear Transformation 3. The Matrix of a Linear Transformation LINEAR TRANSFORMATIONS AND MATRICES 3 DEFINITION - Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L ( u ) in W to each u in V such that: (a) L ( u + v | {z } ∈ V ) = L ( u ) + L ( v ) | {z } ∈ W , for every u and v in V , (b) L ( k u |{z} ∈ V ) = k L ( u ) | {z } ∈ W , for every u in V and every scalar k . In the definition above, observe that • in (a) the + in u + v on the left side of the equation refers to the addition operation in V , whereas the + in L ( u ) + L ( v ) on the right side of the equation refers to the addition operation in W . • Similarly, in (b) the scalar product k u is in V , while the scalar product k L ( u ) is in W . LINEAR TRANSFORMATIONS AND MATRICES 4 As in Lecture Note 13, we shall write the fact that L maps V into W , even if it is not a linear transformation, as L : V → W. If V = W , the linear transformation L : V → V is also called a linear operator on V . LINEAR TRANSFORMATIONS AND MATRICES 5 In Lecture Note 13, we gave a number of examples of linear transformations mapping R n into R m . Thus, the following are linear transformations that we have already discussed: Projection: L : R 3 → R 2 defined by L ( x,y,z ) = ( x,y ) . Dilation: L : R 3 → R 3 defined by L ( u ) = r u , r > 1 . Contraction: L : R 3 → R 3 defined by L ( u ) = r u , < r < 1 . Reflection: L : R 2 → R 2 defined by L ( x,y ) = ( x,- y ) . Rotation: L : R 2 → R 2 defined by L ( u ) =   cos φ- sin φ sin φ cos φ   u . Shear in the x-direction: L : R 2 → R 2 defined by L ( u ) =   1 k 1   u , where k is a scalar. Shear in the y-direction: L : R 2 → R 2 defined by L ( u ) =   1 k 1   u , where k is a scalar. LINEAR TRANSFORMATIONS AND MATRICES 6 Recall that • P 1 is the vector space of all polynomials of degree ≤ 1 ; • in general, P n is the vector space of all polynomials of degree ≤ n , and • M nn is the vector space of all n × n matrices. As in Lecture Note 13, to verify that a given function is a linear transformation, we have to check that conditions (a) and (b) in the preceding definition are satisfied....
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Lecture_14 - Lecture Note 14 Dr Jeff Chak-Fu WONG...

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