Prove that an affine matrix of determinant d changes the area of convex polygons by a...
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Prove that an affine matrix of determinant d changes the area of convex polygons by a factor of d as follows. (a) Show that a matrix A = 1 P 0 0 0 changes the area of a triangle 0 0 1 by a factor of q. Hint: Show that A takes horizontal lines to horizontal lines (b) Show a claim similar to part (a) for a matrix B = T00 S 1 0 0 0 1 (c) Show that an affine matrix M can be written as TAB (or TBA if the center entry of M is 0) where T is a translation, A is a matrix as in part (a), and B is a matrix as in part (b). If e 0, then the matrix M = а b C def 001 can be written as TAB, and if a 0, then it can be written as TBA. You have to write the entries of A and B in terms of the entries of M. That 0 bc leaves the case where the matrix looks like d 0 f for which 0 0 1 no such expression exists, unless I'm mistaken. (d) Use part (c) to extend parts (a) and (b) to any affine matrix. Hint: The determinant of a product is the product of the determinants. For the third case, where a = e = 0, you can use an argument similar to that of Example 4. (e) Extend part (d) from triangles to convex polygons. Prove that an affine matrix of determinant d changes the area of convex polygons by a factor of d as follows. (a) Show that a matrix A = 1 P 0 0 0 changes the area of a triangle 0 0 1 by a factor of q. Hint: Show that A takes horizontal lines to horizontal lines (b) Show a claim similar to part (a) for a matrix B = T00 S 1 0 0 0 1 (c) Show that an affine matrix M can be written as TAB (or TBA if the center entry of M is 0) where T is a translation, A is a matrix as in part (a), and B is a matrix as in part (b). If e 0, then the matrix M = а b C def 001 can be written as TAB, and if a 0, then it can be written as TBA. You have to write the entries of A and B in terms of the entries of M. That 0 bc leaves the case where the matrix looks like d 0 f for which 0 0 1 no such expression exists, unless I'm mistaken. (d) Use part (c) to extend parts (a) and (b) to any affine matrix. Hint: The determinant of a product is the product of the determinants. For the third case, where a = e = 0, you can use an argument similar to that of Example 4. (e) Extend part (d) from triangles to convex polygons.
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