Extra Credit Question 2
Let \(\alpha_1, \dots , \alpha_n\) be columns of a matrix \(A \in \mathbb R^{m \times n}\) and \( R\) the reduced row echelon form of \(A\). Suppose that \(\alpha'_{i_1}, \dots, \alpha'_{i_r}\) are columns of \(R\) which contain leading ones. Show that \(\alpha_{i_1},\dots, \alpha_{i_r}\) forms a basis of the column space of \(A\).Hint: Try to show that \( \alpha_{i_1}, \dots , \alpha_{i_r}\) is a maxmal subset which are linear indepent and consider the equation \[ \sum_{i = 1} ^n x_i \alpha_i = \vec{0}. \]