Given a collection x1, x2, …, xk of vectors in ℝn, we say that the vectors form a linearly dependent set if one of them can be written as a linear combination of the others. That is, the set is linearly dependent if there is an index j in the range 1 to k such that
where the ci’s are given numbers.
The concept of a plane carries over to ℝn: Given a nonzero vector n in ℝn and a point a in ℝn, we define the hyperplane through a perpendicular to n to be all points x ∈ ℝn such that n · (x − a) = 0.
(a) Find the equation for the hyperplane through the point (3, 8, 1, 1, 7) in ℝ5 perpendicular to (1, 0, −1, −3, 2).
(b) Find the equation for the hyperplane in ℝ4 that passes through the points (1, 0, 0, 1), (0, 1, −1, 0), (0, 0, 1, 1), and (0, 0, 0, 2).
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