What are the applications of linear algebra in image processing?

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Are there any linear algebra properties and theorems which when applied to an image produces interesting results?

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One of the most important linear processes in image processing is image convolution—a linear process. Convolution is essential for most algorithms in edge detection and pattern classification through template matching. Another very important application of linear algebra is color conversion, where you transform a pixel's color in one space (e.g., RGB or red-green-blue) to another space (e.g., HSV or hue-saturation-value). Linear operations are central to a number of image compression schemes as well, such as the JPEG compression standard.