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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: Q24E (page 54)

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 $T \overset{鈫�}{(x)} = A \overset{鈫�}{x}$ on this face.
24. $[\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]$

Short Answer

Expert verified

An angle of $90^{鈭�}$ , the face gets rotated counter-clockwise.

Step by step solution

01

Step by Step Explanation: Step 1: Consider the matrix.

Let the matrix be,

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

Then,

$T \overset{鈫�}{(x)} = A \overset{鈫�}{x} T \overset{鈫�}{(x)} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] \overset{鈫�}{x}$

02

Compute the matrix.

The matrix is,

$T \overset{鈫�}{(x)} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] T \overset{鈫�}{(x)} = [\begin{matrix} - x_{2} \\ x_{1} \end{matrix}]$

03

Graph the matrix.

Now, graph the obtained vectors on the given circular face.

$T \overset{鈫�}{(x)}$ is obtained by rotating the vector $\overset{鈫�}{x}$ through an angle of $90^{鈭�}$ in the counter-clockwise direction.

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Most popular questions from this chapter

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

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,

$[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$ ,

Into plane coordinates, $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}]$ . The simplest choice is a linear transformation, for example, the one given by the matrix

$[\begin{matrix} - \frac{1}{2} & 1 & 0 \\ - \frac{1}{2} & 0 & 1 \end{matrix}]$

(a.) Use this transformation to represent the unit cube with corner points

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

Include the images of the axes in your sketch:

(b) Represent the image of the point $[\begin{matrix} 1 \\ \frac{1}{2} \\ \frac{1}{2} \end{matrix}]$ in your figure in part (a)

(c.) Find all the points $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$ in ${鈩�}^{3}$ that are transformed to $[\begin{matrix} 0 \\ 0 \end{matrix}]$ . Explain.

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.

$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] X = I_{2}$

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-1= A?

Consider the transformation T from $R^{2}$ to $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through an angle of $45 掳$ 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 .

See all solutions

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