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Linear map
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===Examples in two dimensions=== In two-[[dimension]]al space '''R'''<sup>2</sup> linear maps are described by 2 × 2 [[matrix (mathematics)|matrices]]. These are some examples: * [[Rotation (mathematics)|rotation]] ** by 90 degrees counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}</math> ** by an angle ''θ'' counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}</math> * [[Reflection (mathematics)|reflection]] ** through the ''x'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}</math> ** through the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}</math> ** through a line making an angle ''θ'' with the origin: <math display="block">\mathbf{A} = \begin{pmatrix}\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}</math> * [[Scaling (geometry)|scaling]] by 2 in all directions: <math display="block">\mathbf{A} = \begin{pmatrix} 2 & 0\\ 0 & 2\end{pmatrix} = 2\mathbf{I}</math> * [[shear mapping|horizontal shear mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & m\\ 0 & 1\end{pmatrix}</math> * skew of the ''y'' axis by an angle ''θ'': <math display="block">\mathbf{A} = \begin{pmatrix} 1 & -\sin\theta\\ 0 & \cos\theta\end{pmatrix}</math> * [[squeeze mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} k & 0\\ 0 & \frac{1}{k}\end{pmatrix}</math> * [[Projection (linear algebra)|projection]] onto the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & 0\\ 0 & 1\end{pmatrix}.</math> If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a [[conformal linear transformation]].
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