If \(C\) is the field of complex numbers, which vectors in \(C^3\) are linear combinations of \((1, 0, -1)\), \((0, 1, 1)\), and \((1, 1, 1)\)?
Solution
Put the three vectors as columns: \[ M= \begin{bmatrix} 1&0&1\\ 0&1&1\\ -1&1&1 \end{bmatrix}. \] Its determinant is \[ \det M=1. \] Hence the columns are linearly independent. Since there are three of them in the three-dimensional space \(\CC^3\), they form a basis. Therefore every vector in \(\CC^3\) is a linear combination of the three given vectors.