An affine transformation $F[\mathbf{x}]$ on $\mathbb{R}^n$ can be expressed in the form $F[\mathbf{x}] = A\mathbf{x} + \mathbf{b}$, where $A$ is an $m \times n$ matrix and $\mathbf{b}$ is a vector in $\mathbb{R}^m$. Here, $A\mathbf{x}$ represents a linear
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