Is the vector \((3, -1, 0, -1)\) in the subspace of \(\RR^4\) spanned by the vectors \((2, -1, 3, 2)\), \((-1, 1, 1, -3)\), and \((1, 1, 9, -5)\)?
Solution
Suppose \[ a(2,-1,3,2)+b(-1,1,1,-3)+c(1,1,9,-5)=(3,-1,0,-1). \] The first three coordinates give \[ 2a-b+c=3, \] \[ -a+b+c=-1, \] \[ 3a+b+9c=0. \] Solving these yields \[ a=1,\qquad b=0,\qquad c=1. \] But then the fourth coordinate of the linear combination is \[ 2a-3b-5c=2-5=-3, \] whereas the target fourth coordinate is \(-1\). This contradiction shows that \((3,-1,0,-1)\) is not in the span of the three given vectors.