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.