a) Let u, v be linearly independent vectors in V2.Show by geometric reasoning that the poi...

a) Let u, v be linearly independent vectors in V2.Show by geometric reasoning that the points P of the plane for which

fill a parallelogram whose edges, properly directed, represent u and v.

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b) With u, v as in (a), let A be a nonsingular 2 × 2 matrix, so that Au, Av are also linearly independent (Problem 13 following Section 1.16) and under the linear mapping y = Ax the parallelogram of part (a) is mapped onto a parallelogram given by

in the plane.

Show that the area of the second parallelogram is |det A| times the area of the first. [Hint: Let B be the matrix whose column vectors are u, v and let C be the matrix whose column vectors are Au, Av. Show that the areas in question are |det B| and |det C|.]