Let A {\displaystyle A} be a 3 × 3 {\displaystyle 3\times 3} matrix, and let S = { v 1 , v 2 , v 3 } {\displaystyle S=\{{\bf {v}}_{1},{\bf {v}}_{2},{\bf {v}}_{3}\}} be a linearly independent set in R 3 {\displaystyle \mathbb {R} ^{3}} such that A v 1 = v 1 , A v 2 = − v 2 , A v 3 = 0 . {\displaystyle A{\bf {v}}_{1}={\bf {v}}_{1},A{\bf {v}}_{2}=-{\bf {v}}_{2},A{\bf {v}}_{3}={\bf {0}}.}
Let w = v 1 + v 2 + v 3 {\displaystyle {\bf {w}}={\bf {v}}_{1}+{\bf {v}}_{2}+{\bf {v}}_{3}} . Show that v 1 , v 2 {\displaystyle {\bf {v}}_{1},\;{\bf {v}}_{2}} and v 3 {\displaystyle {\bf {v}}_{3}} are all in S p a n { w , A w , A 2 w } . {\displaystyle {\rm {Span}}\{{\bf {w}},\;A{\bf {w}},\;A^{2}{\bf {w}}\}.}
be a 
matrix, and let 
be a linearly independent set in 
such that
. Show that 
and 
are all in 
