Engineering Calculus Notes 428

Engineering Calculus Notes 428 - E for more details. When a...

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416 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS several variables. By analogy, we call a mapping L : R 3 R 3 linear if each of its component functions is linear: 2 L ( x,y,z ) = L 1 ( x,y,z ) L 2 ( x,y,z ) . . . L m ( x,y,z ) = a 11 x + a 12 y + a 13 z a 21 x + a 22 y + a 23 z a 31 x + a 32 y + a 33 z . A more eFcient way of writing this is via matrix multiplication : if we form the 3 × 3 matrix [ L ], called the matrix representative of L , whose entries are the coeFcients of the component polynomials [ L ] = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 then the coordinate column of the image L ( −→ x ) is the product of [ L ] with the coordinate column of −→ x : [ L ( −→ x )] = [ L ] · [ −→ x ] or L ( −→ x ) = a 11 x + a 12 y + a 13 z a 21 x + a 22 y + a 23 z a 31 x + a 32 y + a 33 z = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 · x y z . The last equation can be taken as the de±nition of the matrix product; if you are not familiar with matrix multiplication, see Appendix
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Unformatted text preview: E for more details. When a linear mapping is dened in some way other than giving the coordinate polynomials, there is an easy way to nd its matrix representative. The proof of the following is outlined in Exercise 3 : Remark 4.1.1. The j th column of the matrix representative [ L ] of a linear mapping L : R 3 R 3 is the coordinate column of L ( e j ) , where { e 1 , e 2 , e 3 } are the standard basis vectors for R 3 . 2 To avoid tortuous constructions or notations, we will work here with mappings of space to space; the analogues when the domain or target (or both) lives in the plane or on the line are straightforward....
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