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 cross product can be generalized to ℝn. First, let e1 = (1, 0, 0, …, 0), e2= (0, 1, 0, …, 0), e3 = (0, 0, 1, …, 0), …, en = (0, 0, 0, …, 1) be the standard basis vectors for ℝn. The cross product or outer product of vectors a1, a2, a3,…, an−1 in ℝn is defined to be

That is, the basis vectors ei are treated as the entries in the top row of an n ×n determinant, and the components of the vectors ai are the remaining n − 1 rows.

(a) Calculate Ω((4, 1, 3, 0), (−2, 0, 5, 6), (9, 1, 0, 6)).

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(b) Show that Ω(a1, a2, …, an−1) is orthogonal to ai, for i = 1, 2, …, n − 1.

[Hint: Use the fact that ai · Ω = det(ai, a1, …, an−1) and an observation about determinants with two identical rows.]