Linear systems

Throughout this exercise sheet, unless otherwise stated, \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\) and \(n, p\) are positive integers. A linear system of \(n\) equations in \(p\) unknowns is written \(AX = B\) with \(A \in M_{n,p}(\mathbb{K})\), \(X \in M_{p,1}(\mathbb{K})\), \(B \in M_{n,1}(\mathbb{K})\). The augmented matrix is \((A \mid B)\). Elementary row operations are written \(L_i \leftrightarrow L_j\) (swap), \(L_i \leftarrow \lambda L_i\) (dilation, \(\lambda \ne 0\)), \(L_i \leftarrow L_i + \lambda L_j\) (transvection, \(i \ne j\)); the composite shortcut \(L_i \leftarrow \alpha L_i + \beta L_j\) (with \(\alpha \ne 0\), \(i \ne j\)) is the dilation \(L_i \leftarrow \alpha L_i\) followed by the transvection \(L_i \leftarrow L_i + \beta L_j\). The notation \(\mathrm{Vect}(X_1, \ldots, X_r)\) is the set of linear combinations \(\lambda_1 X_1 + \cdots + \lambda_r X_r\) (writing convention, formal theory deferred to Vector spaces). In section 3 (Geometric interpretation), row-operation notation is used as compact shorthand for « replace equation \(i\) by the indicated linear combination »; the formal Gauss algorithm is introduced in section 5.