3. Linear Systems

The linear system

\[\mathbf{A} \mathbf{x} = \mathbf{b}\]

with \(\mathbf{A} \in \mathbb{R}^{m,n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{b} \in \mathbb{R}^m\) has solution if there exists (at least) a vector \(\widetilde{\mathbf{x}}\) whose product with \(\mathbf{A}\) gives \(\mathbf{b}\).

Condition for the existence of a solution. A solution exists if \(\mathbf{b}\) belongs to the range of \(\mathbf{A}\).

Uniqueness of a solution. If a solution \(\widetilde{\mathbf{x}}\) exists, it’s unique if the kernel of \(\mathbf{A}\) is empty, \(K(\mathbf{A}) = \emptyset\). If the kernel is not empty,

\[\mathbf{b} = \mathbf{A}(\widetilde{\mathbf{x}} + \mathbf{u}) = \mathbf{A}\widetilde{\mathbf{x}} + \underbrace{\mathbf{A} \mathbf{u}}_{=\mathbf{0}} \ ,\]

for \(\forall \mathbf{u} \in K(\mathbf{A})\), and thus an infinite number of solutions exists. Given a vector basis of the kernel \(\mathbf{K}(\mathbf{A})\), where \(\text{dim}\left( K(\mathbf{A}) \right) = n_K\), \(\{ \mathbf{u}_1, \dots, \mathbf{u}_{n_K} \}\), the general solution has \(n_K\) “degrees of arbitrarieness”, since the general solution of the problem is

\[\widetilde{\mathbf{x}} + \sum_{i = 1}^{n_K} a_i \mathbf{u}_i = \widetilde{\mathbf{x}} + \mathbf{U} \mathbf{a} \ .\]

todo treat under-, det-, over-determined lin sys