Chapter 4.3, Problem 25E

Let ${\text{v}}_1=\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$, ${\text{v}}_2=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$, ${\text{v}}_3=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, and let H be the set of vectors in ℝ³ whose second and third entries are equal. Then every vector in H has a unique expansion as a linear combination of v₁, v₂, v₃, because $\left[\begin{array}{c}\text{s}\\ \text{t}\\ \text{t}\end{array}\right]$ = $\text{s}\left[\begin{array}{c}\text{1}\\ 0\\ 0\end{array}\right]$ + (t − s) $\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$ + $\text{s}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ for any s and t. Is {v₁, v₂, v₃} a basis for H? Why or why not?