Let \(W\) be the set of all \((x_1, x_2, x_3, x_4, x_5)\) in \(\RR^5\) which satisfy \[ 2x_1 - x_2 + \frac{4}{3}x_3 - x_4 = 0, \] \[ x_1 + \frac{2}{3}x_3 - x_5 = 0, \] \[ 9x_1 - 3x_2 + 6x_3 - 3x_4 - 3x_5 = 0. \] Find a finite set of vectors which spans \(W\).
Solution
Row reduction of the coefficient matrix gives the equivalent equations \[ x_1+\frac23x_3-x_5=0, \qquad x_2+x_4-2x_5=0. \] Thus \(x_3,x_4,x_5\) are free. Writing \[ x_3=s,\qquad x_4=t,\qquad x_5=u, \] we obtain \[ x_1=-\frac23s+u, \qquad x_2=-t+2u. \] Hence every vector in \(W\) has the form \[ s\left(-\frac23,0,1,0,0\right) +t(0,-1,0,1,0) +u(1,2,0,0,1). \] Therefore \(W\) is spanned by \[ \left(-\frac23,0,1,0,0\right), \quad (0,-1,0,1,0), \quad (1,2,0,0,1). \]