Linear functions
We study the properties of linear functions using the deformation of objects.
Discover Bridges
Study Linear functions
LES: Looking at functions
Goal: Solve LES \(\mathbf{Ax}=\mathbf{b}\) (\(\mathbf{A}\in \mathbb{R}^{m\times n}\), \(\mathbf{b}\in\mathbb{R}^m\))
Alternatively: consider function \(f_{\mathbf{A}}: \mathbb{R}^n\rightarrow \mathbb{R}^m\) with \[f_{\mathbf{A}}(\mathbf{x})=\mathbf{Ax}\] and solve \(f_{\mathbf{A}}(\mathbf{x})=\mathbf{b}\). The mentioned function is linear.
Definition 1. Let \(f:\mathbb{R}^n \rightarrow \mathbb{R}^m\) be a function. We say that \(f\) is a linear function if the following properties hold:
For every \(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\): \[f(\mathbf{x}{\color{blue}+}\mathbf{y})=f(\mathbf{x}){\color{blue}+}f(\mathbf{y}).\]
For every \(\mathbf{x}\in\mathbb{R}^n\) and \(\alpha\in\mathbb{R}\): \[f({\color{blue}\alpha} \cdot \mathbf{x})={\color{blue}\alpha \cdot } f(\mathbf{x}).\]
Linear functions map lines onto lines.
Example 2. 
Example 3. 
Geometric operations
Functions of the form \(\mathbf{x} \mapsto \mathbf{Ax}\) can be used to describe geometric operations.
| \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | reflection across \(x_1\)-axis |
|---|---|
| \(\begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix}\) | reflection across \(x_2\)-axis |
| \(\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}\) | stretching by factor \(a\) |
| \(\begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{pmatrix}\) | rotation by the angle \(\alpha\) about the origin |
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The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.