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- Let SCR be a subset. We say S is a hyperplane in R" if there exist an (n − 1)- dimensional subspace WC Rn and a vector v ER" such that S=W+v= {w+v|we W}. Prove the following statements. (a) A subset SC R" is a hyperplane if and only if there exist a₁,..., an, b € R, where a₁,..., an are not all 0, such that S = {(x₁,...,xn) ER" | a₁x₁ + is given by the formula (b) The distance from a point p = (P₁,...,Pn) E Rn to a hyperplane S = {(x₁,...,xn) ER" | a₁x1 + + anxn=b} |a₁p₁ + √a² + + Anxn = + anPn - bl + a²/2 b}.

- Let SCR be a subset. We say S is a hyperplane in R" if there exist an (n − 1)- dimensional subspace WC Rn and a vector v ER" such that S=W+v= {w+v|we W}. Prove the following statements. (a) A subset SC R" is a hyperplane if and only if there exist a₁,..., an, b € R, where a₁,..., an are not all 0, such that S = {(x₁,...,xn) ER" | a₁x₁ + is given by the formula (b) The distance from a point p = (P₁,...,Pn) E Rn to a hyperplane S = {(x₁,...,xn) ER" | a₁x1 + + anxn=b} |a₁p₁ + √a² + + Anxn = + anPn - bl + a²/2 b}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. Let S CRn be a subset. We say S is a hyperplane in R" if there exist an (n − 1)- dimensional subspace WC R" and a vector v ER" such that S=W+v= {w+v|w€ W}. Prove the following statements. (a) A subset SCR" is a hyperplane if and only if there exist a₁,. where a₁,..., an are not all 0, such that S = {(x₁,...,xn) = R" | a₁x₁ + ··· + Anxn = b}. (b) The distance from a point p = (P₁,..., Pn) ER" to a hyperplane S = {(x₁,...,xn) ≤ R" | a₁x₁ + ... + Anxn=b} is given by the formula |a₁p₁ + ● b " an, b ER, + anPn √a² + + a² n (Hint: Do a translation to make S contain the origin, and use the result in 5.)
Transcribed Image Text:6. Let S CRn be a subset. We say S is a hyperplane in R" if there exist an (n − 1)- dimensional subspace WC R" and a vector v ER" such that S=W+v= {w+v|w€ W}. Prove the following statements. (a) A subset SCR" is a hyperplane if and only if there exist a₁,. where a₁,..., an are not all 0, such that S = {(x₁,...,xn) = R" | a₁x₁ + ··· + Anxn = b}. (b) The distance from a point p = (P₁,..., Pn) ER" to a hyperplane S = {(x₁,...,xn) ≤ R" | a₁x₁ + ... + Anxn=b} is given by the formula |a₁p₁ + ● b " an, b ER, + anPn √a² + + a² n (Hint: Do a translation to make S contain the origin, and use the result in 5.)
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