This geometric point of view is obviously useful when we want to model the motion or changes in shape of an object moving in the plane or in 3-space. However, it is useful in higher dimensions as well. The idea that any matrix can be thought of as the product of simpler matrices that correspond to higher dimensional versions of rotation, reflection, projection, shearing, dilation and contraction is of enormous importance to both pure and applied mathematicians. Exercise 1: In general a parallelogram in the plane can be described using vectors. The collection of all points of the form (where p, u and v are vectors and a and b are scalars) will be a parallelogram with vertices at the points p, p + u, p + u + v, and p + v. Find p, u and v for the parallelogram P with vertices at (1,2), (3,3), (4,4), (2,3) -- note that there are four ways to do this. If T is a linear transformation defined by T(x) = Ax, where A is a 2 x 2 matrix, then show that the image T(P) is also a parallelogram by finding its vector description. Show that the vertices of the transformed parallelogram are found by transforming the vertices of the original parallelogram P. A common use of plot is to take a list of data points and graph them, connecting them with straight line segments. For example, The geometric effect of the transformation (u,v) = (x + y, x - y) is clearer if we look at how the unit square transforms. It also helps to look at what happens when we apply the same transformation repeatedly. To make this easier we will use matrices. We begin as before by setting up a vectors x and y that correspond to the vertices of the unit square: NOTE: Be sure to use the plotting option axis equal in the exercises below. This ensures that your disks will be round, not elliptical. Typically, the transformed disk will be elliptical and you want to get a true picture of how the disk changed under the linear transformation. Exercise 2: Let S be the unit square and let T be the transformation T(x,y) = (x - y, x + y) as discussed in the examples above. Plot S, T(S), T(T(S)), and T(T(T(S))) together on the same coordinate axes. Label each figure and summarize your results - that is, explain the geometric effect of the transformation T as clearly as you can. Exercise 3: Consider the linear transformation defined by T(x,y) = (0.8x, 1.4y). What is the geometric effect of this transformation? Plot the disk D together with its transform T(D). Also include the figure you get by applying T to D five times. Label each of the three figures by hand or using the menu bars in your MATLAB plot window to add text to your plot. Again, be sure to use the axis equal option so that disks look like disks, not ellipses. What will the limit figure look like if we continue transforming by T forever? Exercise 8: Consider the basic shearing transformation (x,y) ->, (x + y, y). As we discussed in class, this is a shear parallel to the horizontal axis. What happens if we consider the following sequence of transformations? If we first rotate by 45 degrees the horizontal axis moves into the line y = x. Then apply the shear (x,y) ->, (x+y,y). Finally, rotate in the opposite direction by 45 degrees. What is the combined geometric effect and what is the matrix of the resulting transformation? Exercise 9: A translation is a particularly simple kind of transformation -- for example, a map of the form (x,y) ->, (x + 2, y -3) takes any figure and slides it right by 2 and down by 3. This is not a linear transformation since it cannot be expressed as a matrix A times the vector (x,y). We can use a simple trick called homogeneous coordinates to get around this difficulty if we want to combine translations with rotations or other linear transformations. We simply use (x,y,1) instead of (x,y) to talk about points in the plane. With this change of coordinate system, the translation above becomes (x,y,1) maps to (x + 2, y + 3, 1) and this we can describe this using matrix multiplication: Source.


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