Chapter 3, Problem 8CTB

Given the vectors
${\text{x}}_{\text{1}}=\left(\begin{array}{l}1\\ 2\\ 2\end{array}\right),{\text{x}}_2=\left(\begin{array}{l}1\\ 3\\ 3\end{array}\right)$ ,
${\text{x}}_3=\left(\begin{array}{l}1\\ 5\\ 5\end{array}\right),{\text{x}}_4=\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)$
(a) Are ${\text{x}}_1{\text{,x}}_2{\text{,x}}_3$ , and ${\text{x}}_4$ linearly independent in ${ℝ}^{\text{3}}$ ? Explain.
(b) Do ${\text{x}}_1{\text{,x}}_2$ span ${ℝ}^{\text{3}}$ ? Explain.
(c) Do ${\text{x}}_1{\text{,x}}_2{\text{,x}}_3$ , span ${ℝ}^{\text{3}}$ ? Are they linearly independent? Do they form a basis for ${ℝ}^{\text{3}}$ ? Explain.
(d) Do ${\text{x}}_1{\text{,x}}_2{\text{,x}}_4$ span ${ℝ}^{\text{3}}$ ? Are they linearly independent? Do they form a basis for ${ℝ}^{\text{3}}$ ? Explain or prove your answers.