. (a) A hyperplane in the n-dimensional Euclidean space Rn has an equation of the form a1x1 + a2x2 + a3x3 ++ anxn + an+1 = 0 where ai, i = 1, 2, 3,..., n + 1, are constants, not all zero, and xi, i = 1, 2, 3,...,n, are variables for which (x1, x2, x3,...,xn) Rn A point (x10, x20, x30,...,xn0) Rn lies on this hyperplane if a1x10 + a2x20 + a3x30 ++ anxn0 + an+1 = 0 Given that the n points(x1i, x2i, x3i,...,xni), i = 1, 2, 3,..., n, lie on this hyperplane and that they uniquely determine the equation of the hyperplane, show that the equation of the hyperplane can be written in determinant form as x1 x2 x3 xn 1 x11 x21 x31 xn1 1 x12 x22 x32 xn2 1 x13 x23 x33 xn3 1 . . . . . . . . . ... . . . . . . x1n x2n x3n xnn 1 = 0 (b) Determine the equation of the hyperplane in R9 that goes through the following nine points: (1, 2, 3, 4, 5, 6, 7, 8, 9) (2, 3, 4, 5, 6, 7, 8, 9, 1) (3, 4, 5, 6, 7, 8, 9, 1, 2) (4, 5, 6, 7, 8, 9, 1, 2, 3) (5, 6, 7, 8, 9, 1, 2, 3, 4) (6, 7, 8, 9, 1, 2, 3, 4, 5) (7, 8, 9, 1, 2, 3, 4, 5, 6) (8, 9, 1, 2, 3, 4, 5, 6, 7) (9, 1, 2, 3, 4, 5, 6, 7, 8)

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