(Solution) - Let the columns of be the homogeneous form of the coordinates -(2025 Original AI-Free Solution)

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Let the columns of

be the homogeneous form of the coordinates of the vertices of a triangle in the plane. Note that the first and last columns are the same, indicating that the figure is a closed region.
(a) In a coordinate plane, sketch the triangle determined by S. Connect its vertices with straight line segments.
(b) The triangle determined by 5 is to be scaled by ^ in both the x- and y-directions and then translated by
Determine the 3 x 3 matrix M in homogeneous form that represents the composite transformation that first performs the scaling and then the translation.
(c) Use the matrix M from part (b) to determine the image of the triangle and sketch it in a coordinate plane.
(d) Determine the matrix Q in homogeneous form that would first perform the translation and then the scaling.
(e) Use the matrix Q from part (d) to determine the image of the triangle and sketch it in a coordinate plane.
(f) Are the images from parts (c) and (e) the same? Explain why or why not.