Let B={v1,v2,…,vm} be a basis for Rm. Suppose kvm is a linear combination of v1,v2,⋯,vm−1 for some scalar k. What can be said about the possible value(s) of k?

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CoormaBak9

Answered question

2021-01-31

Let $B = {v 1, v 2, \dots, v m}$ be a basis for $R^{m}$ . Suppose kvm is a linear combination of $v 1, v 2, \dots, v m - 1$ for some scalar k. What can be said about the possible value(s) of k?

Answer & Explanation

Corben Pittman

Skilled2021-02-01Added 83 answers

k must be 0. For B to be a basis for $R^{m}$ , it must be linearly independent. Scaling a set of vectors by a non-zero number has no effect on whether they are linearly independent or not because the directions don't change when you scale vectors.
However, when you scale a vector by the number 00, then you end up with the zero vector. And the zero vector is a linear combination of any collection of vectors you want. In particular,
$0 v m = 0 = 0 v 1 + 0 v 2 + \dots + 0 v m^{- 1}$
You'll likely need to prove or find the theorems from your book/ class that prove some of the statements I've made here (or really just that third sentence), but that's the general idea.

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