Show that the vectors \[ \alpha_1 = (1, 1, 0, 0), \quad \alpha_2 = (0, 0, 1, 1), \] \[ \alpha_3 = (1, 0, 0, 4), \quad \alpha_4 = (0, 0, 0, 2) \] form a basis for \(\RR^4\). Find the coordinates of each of the standard basis vectors in the ordered basis \(\{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}\).
Solution
Put the four vectors as columns of \[ A= \begin{bmatrix} 1&0&1&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&1&4&2 \end{bmatrix}. \] Its determinant is \(2\ne0\), so the columns form a basis of \(\RR^4\).
The inverse matrix is \[ A^{-1}= \begin{bmatrix} 0&1&0&0\\ 0&0&1&0\\ 1&-1&0&0\\ -2&2&-\frac12&\frac12 \end{bmatrix}. \] The \(j\)th column of \(A^{-1}\) is the coordinate vector of the standard basis vector \(e_j\) in the ordered basis \(\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}\). Thus \[ [e_1]_{\mathfrak B}=(0,0,1,-2)^t, \] \[ [e_2]_{\mathfrak B}=(1,0,-1,2)^t, \] \[ [e_3]_{\mathfrak B}=(0,1,0,-\tfrac12)^t, \] \[ [e_4]_{\mathfrak B}=(0,0,0,\tfrac12)^t. \]