(Solution) - Show that linear maps preserve the linear structures of a -(2025 Original AI-Free Solution)

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Show that linear maps preserve the linear structures of a space.
(a) Show that for any linear map from Rn to Rm, the image of any line is a line. The image may be a degenerate line, that is, a single point.
(b) Show that the image of any linear surface is a linear surface. This generalizes the result that under a linear map the image of a subspace is a subspace.
(c) Linear maps preserve other linear ideas. Show that linear maps preserve "betweeness": if the point B is between A and C then the image of B is between the image of A and the image of C.