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Chapter 2: Q33E (page 54)
Consider the transformation T from to that rotates any vectorthrough an angle of in the counterclockwise direction, as shown in the following figure:
You are told that T is a linear transformation. (This will be shown in the next section.) Find the matrix of T .
Matrix for T will be .
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When you represent a three 鈥揹imensional object graphically in the plane (on paper, the blackboard, or a computer screen), you have to transform spatial coordinates,
,
Into plane coordinates, . The simplest choice is a linear transformation, for example, the one given by the matrix
(a.) Use this transformation to represent the unit cube with corner points
Include the images of the axes in your sketch:
(b) Represent the image of the point in your figure in part (a)
(c.) Find all the points in that are transformed to . Explain.
Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation on this face.
24.
Use the given partitions to compute the products in Exercises 15 and 16. Check your work by computing the same products without using a partition. Show all your work.
16.
There exists a nonzero upper triangular 2 脳 2 matrix A such that .
One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about a line. Identify both and briefly justify your choice.
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