Given a collection x1, x2, …, xk of vectors in ℝn, we say that the vectors form a linearly...

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 line carries over to ℝn : Given a direction vector m in ℝn and a point a in ℝn, we define the line through a with direction m to be the set of points x given by

 x = tm + a,

where t ranges over all real numbers. We call this a parametrization of the line.

(a) Find a parametrization of the line in ℝ6 that passes through the points (8, −7, 3, −4, 0, 9) and (3, 0, 8, 0, 13, 1).

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(b) Find a parametrization for the line through (−1, −1, −1, −1) that is perpendicular to the hyperplane 7x1 − 3x2 + 4x3 + x4 = 5 in ℝ4.