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

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?

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.
$$\displaystyle{0}{v}{m}={0}={0}{v}{1}+{0}{v}{2}+⋯+{0}{v}{m}^{{−{1}}}$$