Chapter 3, Problem 13CTA

Answer each of the statements that follows as true or false. In each case, explain or prove your answer.

13. Let ${\text{x}}_{\text{1}},{\text{x}}_{\text{2}},\mathrm{...},{\text{x}}_k$ be linearly independent vectors in ${ℝ}^n$ . If $k<n$ and ${\text{x}}_{k+1}$ is a vector that is not in Span $\left({\text{x}}_{\text{1}},{\text{x}}_{\text{2}},\mathrm{...},{\text{x}}_k\right)$ , then the vectors ${\text{x}}_{\text{1}},{\text{x}}_{\text{2}},\mathrm{...},{\text{x}}_k$ , ${\text{x}}_{k+1}$ are linearly independent.