Complete the proof of the formula for the k-volume of a
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Complete the proof of the formula for the k-volume of a k-parallelogram spanned by the vectors v1, ::: , vk by induction
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Complete the proof of the formula for the k-volume of a k-parallelogram spanned by the vectors v1, ::: , vk by induction on k. The case k D 1 is trivial, and the case k D 2 is done in the previous exercise. A .k C 1/-parallelogram has 2.k C 1/ faces, each of which is a k-parallelogram. The .k C 1/-volume of the .k C 1/-parallelogram is the k-volume of one of its faces (say, spanned by v1, ::: , vk) multiplied by the length h of the perpendicular projection of the remaining edge vkC1 onto the k-dimensional subspace containing that face. You will find Cramer
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