Find the coordinate matrix of the vector \((1, 0, 1)\) in the basis of \(C^3\) consisting of the vectors \((2i, 1, 0)\), \((2, -1, 1)\), \((0, 1+i, 1-i)\), in that order.
Solution
Seek scalars \(a,b,c\in\CC\) such that \[ a(2i,1,0)+b(2,-1,1)+c(0,1+i,1-i)=(1,0,1). \] Solving the resulting system gives \[ a=2-i, \qquad b=-\frac12-\frac{3i}{2}, \qquad c=-1+\frac{i}{2}. \] Therefore the coordinate matrix of \((1,0,1)\) in the given ordered basis is \[ \begin{bmatrix} 2-i\\ -\frac12-\frac{3i}{2}\\ -1+\frac{i}{2} \end{bmatrix}. \]