Linear Algebra by otto brestscher chapter 01

Linear Algebra with Applications (3rd Edition)

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SSM: Linear Algebra Section 1.1 Chapter 1 1.1 1. ¯ ¯ ¯ ¯ x +2 y =1 2 x +3 y ¯ ¯ ¯ ¯ 2 × 1st equation ¯ ¯ ¯ ¯ x y y = 1 ¯ ¯ ¯ ¯ ÷ ( 1) ¯ ¯ ¯ ¯ x y y ¯ ¯ ¯ ¯ 2 × 2nd equation ¯ ¯ ¯ ¯ x = 1 y ¯ ¯ ¯ ¯ ,sothat( x,y )=( 1 , 1). 3. ¯ ¯ ¯ ¯ 2 x +4 y =3 3 x +6 y =2 ¯ ¯ ¯ ¯ ÷ 2 ¯ ¯ ¯ ¯ x y = 3 2 3 x y ¯ ¯ ¯ ¯ 3 × 1st equation ¯ ¯ ¯ ¯ x y = 3 2 0= 5 2 ¯ ¯ ¯ ¯ So there is no solution. 5. ¯ ¯ ¯ ¯ 2 x y =0 4 x +5 y ¯ ¯ ¯ ¯ ÷ 2 ¯ ¯ ¯ ¯ x + 3 2 y 4 x y ¯ ¯ ¯ ¯ 4 × 1st equation ¯ ¯ ¯ ¯ x + 3 2 y y ¯ ¯ ¯ ¯ ÷ ( 1) ¯ ¯ ¯ ¯ x + 3 2 y y ¯ ¯ ¯ ¯ 3 2 × 2nd equation ¯ ¯ ¯ ¯ x y ¯ ¯ ¯ ¯ , so that ( )=(0 , 0). 7. ¯ ¯ ¯ ¯ ¯ ¯ x y z x y z x y z =4 ¯ ¯ ¯ ¯ ¯ ¯ I I ¯ ¯ ¯ ¯ ¯ ¯ x y z y + z 2 y z ¯ ¯ ¯ ¯ ¯ ¯ 2( II ) 2( ) ¯ ¯ ¯ ¯ ¯ ¯ x + z = 3 y + z 1 ¯ ¯ ¯ ¯ ¯ ¯ This system has no solution. 9. ¯ ¯ ¯ ¯ ¯ ¯ x y z 3 x y + z 7 x y 3 z ¯ ¯ ¯ ¯ ¯ ¯ 3( I ) 7( I ) ¯ ¯ ¯ ¯ ¯ ¯ x y z 4 y 8 z = 2 12 y 24 z = 6 ¯ ¯ ¯ ¯ ¯ ¯ ÷ ( 4) ¯ ¯ ¯ ¯ ¯ ¯ x y z y z = 1 2 12 y 24 z = 6 ¯ ¯ ¯ ¯ ¯ ¯ 2( ) +12( ) ¯ ¯ ¯ ¯ ¯ ¯ x z y z = 1 2 0 ¯ ¯ ¯ ¯ ¯ ¯ This system has infnitely many solutions: iF we choose z = t , an arbitrary real number, then we get x = z = t and y = 1 2 2 z = 1 2 2 t . ThereFore, the general solution is ( x,y,z )= ( t, 1 2 2 t,t ) , where t is an arbitrary real number. 11. ¯ ¯ ¯ ¯ x 2 y 3 x y =17 ¯ ¯ ¯ ¯ 3( I ) ¯ ¯ ¯ ¯ x 2 y 11 y =11 ¯ ¯ ¯ ¯ ÷ 11 ¯ ¯ ¯ ¯ x 2 y y ¯ ¯ ¯ ¯ +2( ) ¯ ¯ ¯ ¯ x y ¯ ¯ ¯ ¯ , so that ( )=(4 , 1). See ±igure 1.1. 1
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Chapter 1 SSM: Linear Algebra Figure 1.1: for Problem 1.1.11 . Figure 1.2: for Problem 1.1.13 . 13. ¯ ¯ ¯ ¯ x 2 y =3 2 x 4 y =8 ¯ ¯ ¯ ¯ 2( I ) ¯ ¯ ¯ ¯ x 2 y 0= 2 ¯ ¯ ¯ ¯ , which has no solutions. (See Figure 1.2.) 15. The system reduces to ¯ ¯ ¯ ¯ ¯ ¯ x =0 y z ¯ ¯ ¯ ¯ ¯ ¯ so the unique solution is ( x,y,z )=(0 , 0 , 0). The three planes intersect at the origin. 17. ¯ ¯ ¯ ¯ x +2 y = a 3 x +5 y = b ¯ ¯ ¯ ¯ 3( I ) ¯ ¯ ¯ ¯ x y = a y = 3 a + b ¯ ¯ ¯ ¯ ÷ ( 1) ¯ ¯ ¯ ¯ x y = a y a b ¯ ¯ ¯ ¯ 2( II ) ¯ ¯ ¯ ¯ x = 5 a b y a b ¯ ¯ ¯ ¯ ,sothat( x,y )=( 5 a b, 3 a b ). 19. a. Note that the demand D 1 for product 1 increases with the increase of price P 2 ; likewise the demand D 2 for product 2 increases with the increase of price P 1 . This indicates that the two products are competing; some people will switch if one of the products gets more expensive. 2
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SSM: Linear Algebra Section 1.1 b. Setting D 1 = S 1 and D 2 = S 2 we obtain the system ¯ ¯ ¯ ¯ 70 2 P 1 + P 2 = 14 + 3 P 1 105 + P 1 P 2 = 7+2 P 2 ¯ ¯ ¯ ¯ , or ¯ ¯ ¯ ¯ 5 P 1 + P 2 = 84 P 1 3 P 2 = 112 ¯ ¯ ¯ ¯ , which yields the unique solution P 1 =26and P 2 = 46.
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Linear Algebra by otto brestscher chapter 01 - SSM: Linear...

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