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Chapter 2: Q29E (page 97)
For which values of the constant k is the following matrix invertible?
For each kdistinct than 1 or 2.
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Find the matrices of the linear transformations from R3to R3given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
19. The orthogonal projection onto the x-y-plane.
Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16through 23 . Show the effect of these transformations on the letter considered in Example 5 . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.
21.
Consider the regular tetrahedron sketched below, whose center is at the
origin.
Let T from to be the rotation about the axis through the points 0 and that transforms into . Find the images of the four corners of the tetrahedron under this transformation.
Let L from to be the reflection about the plane through the points 0 , . Find the images of the four corners of the tetrahedron under this transformation.
Describe the transformations in parts (a) through (c) geometrically.
d. Find the images of the four corners under the transformations . Are the two transformations the same?
e. Find the images of the four corners under the transformation . Describe this transformation geometrically
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.
Give a geometric interpretation of the linear transformations defined by the matrices in Exercisesthrough . Show the effect of these transformations on the letter considered in Example. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise
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