Prove that if $\{u_1 ,\dots , u_k, v\}$ is linearly dependent then $v\in\text{span}\{u_1 ,\dots , u_k\}$

Question

Let F be a field. Suppose $\{u_1 ,\dots , u_k\}\subseteq F^n$ is a linearly independent set, and $v \in F^n$ does not belong to this set. Prove that if $\{u_1 ,\dots , u_k, v\}$ is linearly dependent then $v\in\text{span}\{u_1 ,\dots , u_k\}$.

Answer 1

We can take the definition of $\{u_1,\dots,u_k,v\}$ being linearly dependent to mean that there must exist $\lambda_i$ (with $i\in\{1,\dots,k+1\}$) not all zero such that: $$ \lambda_1u_1+\dots+\lambda_ku_k+\lambda_{k+1}v=0 $$ Then if we suppose that $\lambda_{k+1}=0$ it's clear this would mean that $\{u_1,\dots,u_k\}$ must be linearly dependent, but this is contrary to our assumption. So we must then conclude that $\lambda_{k+1}\neq 0$ which would mean: $$ v=-\frac{\lambda_1}{\lambda_{k+1}}u_1+\dots+\left(-\frac{\lambda_k}{\lambda_{k+1}}\right)u_k $$ so that: $$ v\in \text{span}\{u_1,\dots,u_k\} $$