Question

linear algebra

Remember we were able to express rotations and reflections, which are geometric transformations, using a linear transformation T, the coef- ficient matrix corresponding to the geometric transformation (r. y) (r', ) (a) What problem do you encounter with translations (r. y) (r+ h.y+k)? To handle this problem, We let the vector (x, y1 ) in R2 correspond to the vector (x1, y1, 1), and conversely. (In effect, we're projecting the :xy-plane onto the plane 1) introduce homogeneous coordinates. we (b) Find M, the matrix corresponding to the translation (r, y, 1) - (rh, y+k, 1). In other words, find M so that y+k 1 1 1 a12 a13 (c) Let T = Consider the square with opposite ver- a21 a22 a23 1 0 0 tices at (0,0) and (1, 1). Find where each of the four vertices of the square ends up when the linear transformation M is applied to each of them. (d) Find the area of the resulting quadrilateral. (e) Suppose that when T is applied to a unit square, the transformed square has area 0. Does T 0 (the 3 x 3 matrix of all zeroes)? Prove, or give a counter-example.

Answer 1

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