Question

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

Vectors and spaces
ANSWERED
asked 2021-01-31

Let \(\displaystyle{B}={\left\lbrace{v}{1},{v}{2},\ldots,{v}{m}\right\rbrace}\) be a basis for \(R^{m}\). Suppose kvm is a linear combination of \(v1, v2, \cdots, vm-1\) for some scalar k. What can be said about the possible value(s) of k?

Expert Answers (1)

2021-02-01

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,
\(\displaystyle{0}{v}{m}={0}={0}{v}{1}+{0}{v}{2}+⋯+{0}{v}{m}^{{−{1}}}\)
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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