MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 3 a Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) to the second view (with the front view of the green letter C). Call it A. b Write the standard matrix of a linear transformation that transforms the second view (with the front view of the green letter C) to the third view (with the front view of the purple letter C). Call it B c Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) directly to the third view (with the front view of the purple letter C). Call it D. What is the relationship between this matrix and the previous ones? You notice a symmetry in these three views of the solved puzzle. You remember the three pieces of the puzzle were identical in shape. That makes you think that the position of the pieces in the solved puzzle are interchangeable. That is: where the red piece lies now in Figure [0.1, could be lying a purple piece. So you decide to find a transformation that when applied to the purple piece in Figure 0.1 moves it to the position of the red piece in Figure 0.1. (1) Let p be the position vector of the point p marked in Figure 0.1. Find a vector 5 such that p+b=0. (2) Consider the transformation F(x) = x+b, where b is the vector you found in the previous part. What is F(0)? Is F a linear transformation? Justify. (3) Let B denote the set of vectors whose tips when in standard position, create the purple piece in Figure 0.1. You don't need to describe B in set notation (even though you can if you wish). Sketch a picture (by hand or your choice of software) that shows {F(x) | € B}, that is, the image of B under F. Mark the origin and the axis on your sketch. Mark and find the coordinates of all the corners of the front of the purple letter C. (4) Compare you image with this one. Is it the same? The other pieces are moved away to make room for the purple piece. У FIGURE 0.2. The purple piece translated (5) Find two linear transformations T and S such that To S moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. Describe T and S via their standard matrices. (6) Find a single linear transformation that moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. (7) Find a transformation that moves the purple piece in Figure 0.1 to the position of the red piece in Figure 0.1. Is this a linear transformation? Justify. MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 3 a Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) to the second view (with the front view of the green letter C). Call it A. b Write the standard matrix of a linear transformation that transforms the second view (with the front view of the green letter C) to the third view (with the front view of the purple letter C). Call it B c Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) directly to the third view (with the front view of the purple letter C). Call it D. What is the relationship between this matrix and the previous ones? You notice a symmetry in these three views of the solved puzzle. You remember the three pieces of the puzzle were identical in shape. That makes you think that the position of the pieces in the solved puzzle are interchangeable. That is: where the red piece lies now in Figure [0.1, could be lying a purple piece. So you decide to find a transformation that when applied to the purple piece in Figure 0.1 moves it to the position of the red piece in Figure 0.1. (1) Let p be the position vector of the point p marked in Figure 0.1. Find a vector 5 such that p+b=0. (2) Consider the transformation F(x) = x+b, where b is the vector you found in the previous part. What is F(0)? Is F a linear transformation? Justify. (3) Let B denote the set of vectors whose tips when in standard position, create the purple piece in Figure 0.1. You don't need to describe B in set notation (even though you can if you wish). Sketch a picture (by hand or your choice of software) that shows {F(x) | € B}, that is, the image of B under F. Mark the origin and the axis on your sketch. Mark and find the coordinates of all the corners of the front of the purple letter C. (4) Compare you image with this one. Is it the same? The other pieces are moved away to make room for the purple piece. У FIGURE 0.2. The purple piece translated (5) Find two linear transformations T and S such that To S moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. Describe T and S via their standard matrices. (6) Find a single linear transformation that moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. (7) Find a transformation that moves the purple piece in Figure 0.1 to the position of the red piece in Figure 0.1. Is this a linear transformation? Justify.