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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q37E (page 73)

The trace of a matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ is the sum $a + d$ of its diagonal entries. What can you say about the trace of a localid="1664169930199" $2 脳 2$ matrixthat represents localid="1664169941875" $a (n)$ .
a. orthogonal projection b. reflection about a line
c. rotation d. (horizontal or vertical) shear.
In three cases, give the exact value of the trace, and in one case, give an interval of possible values.

Short Answer

Expert verified

a. The orthogonal projection is, 1.

b. The reflection about a line is, 0.

c. The rotation is, $- 2 鈮� t r (R) 鈮� 2$

d. The horizontal or vertical shear is, 2.

Step by step solution

01

Compute the orthogonal projection.

(a)

Consider the matrix,

$a (n) = [\begin{matrix} a^{2} & a b \\ a b & b^{2} \end{matrix}]$

The trace of the matrix is,

$a^{2} + b^{2} = 1$

Where, $[\begin{matrix} a \\ b \end{matrix}]$ is a unit vector.

Hence, the orthogonal projection is 1.

02

Compute the reflection about a line.

(b)

Consider the matrix,

$A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

The reflection matrix is look like, $[\begin{matrix} a & b \\ b & - a \end{matrix}]$

The trace of matrix is,

$a - a = 0$

Hence, the trace of the matrix is 0.

03

Compute the rotation

(c)

Consider the matrix,

$R = [\begin{matrix} \cos 胃 & - \sin 胃 \\ \sin 胃 & \cos 胃 \end{matrix}]$

The rotation matrix is of form,

$\begin{array}{rcl} \cos 胃 + \cos 胃 & = & 2 \cos 胃 \\ 鈬� & - 1 鈮� \cos 胃鈮� 1 \\ 鈬� & - 2 鈮� 2 \cos 胃鈮� 2 \end{array}$

The trace of the matrix is,

$- 2 鈮� t r (R) 鈮� 2$

04

Compute the horizontal/vertical shear.

(d)

Consider the matrix,

$K = [\begin{matrix} 1 & K \\ 0 & 1 \end{matrix}] or [\begin{matrix} 1 & 0 \\ K & 1 \end{matrix}]$

The trace of matrix is,

$1 + 1 = 2$

Hence, the horizontal or vertical shear is,2.

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Most popular questions from this chapter

Question:If A is an invertible matrix andcis a nonzero scalar, is the matrixcA invertible? If so, what is the relationship betweenA^-1 and( cA )^-1?

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ onto line L.

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises $16$ through $23$ . Show the effect of these transformations on the letter $L$ 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 $13$

$[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$

Consider the regular tetrahedron sketched below, whose center is at the

origin.

Let T from ${鈩�}^{3}$ to ${鈩�}^{3}$ be the rotation about the axis through the points 0 and $P_{2}$ that transforms $P_{1}$ into $P_{3}$ . Find the images of the four corners of the tetrahedron under this transformation.

$P_{0} \overset{r}{鈫�} P_{1} \underset{}{鈫�} P_{3} P_{2} \underset{}{鈫�} P_{3} \underset{}{鈫�}$

Let L from ${鈩�}^{3}$ to ${鈩�}^{3}$ be the reflection about the plane through the points 0 , $p_{0} and p_{3}$ . Find the images of the four corners of the tetrahedron under this transformation.

$P_{0} \overset{L}{鈫�} P_{1} \overset{}{鈫�} P_{2} \overset{}{鈫�} P_{3} \overset{}{鈫�}$

Describe the transformations in parts (a) through (c) geometrically.

$a . T^{- 1} b . L^{- 1} c . T^{2} = T 鈭� T 鈥� (t h e c o m p o s i t e o f T w i t h i t s e l f)$

d. Find the images of the four corners under the transformations $T 鈭� L a n d L 鈭� T$ . Are the two transformations the same?

$P_{0} \overset{T 鈭� L}{鈫�} P_{0} \overset{T 鈭� L}{鈫�} P_{1} \overset{}{鈫�} P_{1} \overset{}{鈫�} P_{2} \overset{}{鈫�} P_{2} \overset{}{鈫�} P_{3} \overset{}{鈫�} P_{3} \overset{}{鈫�}$

e. Find the images of the four corners under the transformation $L 鈭� T 鈭� L$ . Describe this transformation geometrically

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 $L$ 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. $[\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$

See all solutions

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