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 efficient 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 coefficients 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 definition 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 de±ned 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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