Let S CR be a subset. We say S is a hyperplane in R if there...

 

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Let S CR be a subset. We say S is a hyperplane in R" if there exist an (n - 1)- dimensional subspace W CR" and a vector v ER such that S=W+v={w+v|wЄW}. Prove the following statements. (a) A subset S CR is a hyperplane if and only if there exist a₁,..., an, bЄ R, where a₁,..., an are not all 0, such that S={(x1,xn) ЄR" | a1x1 + +anxn = b}. (b) The distance from a point p = (P1,..., Pn) E R" to a hyperplane S={(x1,...,xn) ЄR" | a1x1 + +anxn = b} is given by the formula |a1p1 + +anpn - b √√a²+...+a² (Hint: Do a translation to make S contain the origin, and use the result in 5.) Let S CR be a subset. We say S is a hyperplane in R" if there exist an (n - 1)- dimensional subspace W CR" and a vector v ER such that S=W+v={w+v|wЄW}. Prove the following statements. (a) A subset S CR is a hyperplane if and only if there exist a₁,..., an, bЄ R, where a₁,..., an are not all 0, such that S={(x1,xn) ЄR" | a1x1 + +anxn = b}. (b) The distance from a point p = (P1,..., Pn) E R" to a hyperplane S={(x1,...,xn) ЄR" | a1x1 + +anxn = b} is given by the formula |a1p1 + +anpn - b √√a²+...+a² (Hint: Do a translation to make S contain the origin, and use the result in 5.)

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