(Based on 2.1 Problem (6) Vector addition and scalar multiplication are required to satisfy these eight...

 

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(Based on 2.1 Problem (6) Vector addition and scalar multiplication are required to satisfy these eight rules: Rule 1: x+y=y+x (commutative property) Rule 2: x+(y+z)=(x+y)+z (associative property) Rule 3: There is a single unique "zero vector" such that x + 0 = x for all ï (additive identity property) Rule 4: For each x there is a unique vector −ï such that x + (−x) = 0 (existence of additive inverses) Rule 5: There is a unique identity element “1” such that 1x = x (multiplicative identity property) Rule 6: (c₁c₂)x= c₁(c₂x) (associative property - scalar multiplication) Rule 7: c(x + y) = cx + cy (distributive property - vector addition) Rule 8: (C₁+C₂)x= c₁x + c₂c (distributive property - scalar addition) (a) Suppose addition in R² adds an extra 1 to each component, so that (3,1) + (5,0) (3 +5+1,1+0+ 1) = (9, 2) instead of (8,1) and in general à + y = (x₁, x2) + (y₁, Y2) (x₁+Y₁+1, x2 + y2 + 1). With scalar multiplication unchanged, which of the above eight rules are broken? Why? (b) Show that the set of all positive real numbers with x + y redefined to ry and cx redefined to x is a vector space (i.e. show that if x, y € V then xy € V and if x = V then xº € V for all real numbers c). What is the "zero vector? (c) Suppose (x1, x2)+(y₁, y2) is defined to be (x₁+y2, x2+y₁). With the usual cx = = (cx1, cx2), which of the eight conditions are not satisfied? (Based on 2.1 Problem (6) Vector addition and scalar multiplication are required to satisfy these eight rules: Rule 1: x+y=y+x (commutative property) Rule 2: x+(y+z)=(x+y)+z (associative property) Rule 3: There is a single unique "zero vector" such that x + 0 = x for all ï (additive identity property) Rule 4: For each x there is a unique vector −ï such that x + (−x) = 0 (existence of additive inverses) Rule 5: There is a unique identity element “1” such that 1x = x (multiplicative identity property) Rule 6: (c₁c₂)x= c₁(c₂x) (associative property - scalar multiplication) Rule 7: c(x + y) = cx + cy (distributive property - vector addition) Rule 8: (C₁+C₂)x= c₁x + c₂c (distributive property - scalar addition) (a) Suppose addition in R² adds an extra 1 to each component, so that (3,1) + (5,0) (3 +5+1,1+0+ 1) = (9, 2) instead of (8,1) and in general à + y = (x₁, x2) + (y₁, Y2) (x₁+Y₁+1, x2 + y2 + 1). With scalar multiplication unchanged, which of the above eight rules are broken? Why? (b) Show that the set of all positive real numbers with x + y redefined to ry and cx redefined to x is a vector space (i.e. show that if x, y € V then xy € V and if x = V then xº € V for all real numbers c). What is the "zero vector? (c) Suppose (x1, x2)+(y₁, y2) is defined to be (x₁+y2, x2+y₁). With the usual cx = = (cx1, cx2), which of the eight conditions are not satisfied?