?Prove part of Theorem 8.6.1. [Hint: A line in the plane has an equation of the form

Use the method of Example 1 to find an equation for the image of the line \(y=4 x\) under multiplication by the matrix \(A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 1 \end{array}\right]\) Equation Transcription: [] Text Transcription: y=4x A=[ 5 2 \ 2 1 ]

Use the method of Example 1 to find an equation for the image of the line \(y=-4 x+3\) under multiplication by the matrix \(A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 1 \end{array}\right]\) Equation Transcription: [] Text Transcription: y=-4x+3 A=[ 5 2 \ 2 1 ]

In Exercises 3-4, find an equation for the image of the line \(y=2 x\) that results from the stated transformation. A shear by a factor 3 in the \(x\) -direction. Equation Transcription: Text Transcription: y=2x x

In Exercises 3-4, find an equation for the image of the line \(y=2 x\) that results from the stated transformation. A compression with factor \(\frac{1}{2}\) in the \(y\) -direction. Equation Transcription: Text Transcription: y=2x 1\2 y

In Exercises 5–6, sketch the image of the unit square under multiplication by the given invertible matrix. As in Example 2, number the edges of the unit square and its image so it is clear how those edges correspond. \(\left[\begin{array}{ll} 3 & -1 \\ 1 & -2 \end{array}\right]\) Equation Transcription: [] Text Transcription: [ 3 -1 \ 1 -2 ]

In Exercises 5–6, sketch the image of the unit square under multiplication by the given invertible matrix. As in Example 2, number the edges of the unit square and its image so it is clear how those edges correspond. \(\left[\begin{array}{rr} 2 & 1 \\ -1 & 2 \end{array}\right]\) Equation Transcription: [] Text Transcription: [ 2 1 \-1 2 ]

In each part of Exercises , find the standard matrix for a single operator that performs the stated succession of operations. a. Compresses by a factor of \(\frac{1}{2}\) in the \(x\) -direction, then expands by a factor of 5 in the \(y\)-direction. b. Expands by a factor of 5 in the \(y\) -direction, then shears by a factor of 2 in the \(y\) -direction. c. Reflects about \(y=x\), then rotates through an angle of \(180^{\circ}\) about the origin. Equation Transcription: Text Transcription: 1\2 x y y y y=x 180°

In each part of Exercises , find the standard matrix for a single operator that performs the stated succession of operations. a. Reflects about the \(y-\text { axis }\), then expands by a factor of 5 in the \(x\) -direction, and then reflects about \(y=x\). b. Rotates through \(30^{\circ}\) about the origin, then shears by a factor of \(-2\) in the \(y\) -direction, and then expands by a factor of 3 in the \(y\) -direction. Equation Transcription: Text Transcription: y-axis x y=x 30° -2 y y

In each part of Exercises 9-10, determine whether the stated operators commute. a. A reflection about the \(x-a x i s\) and a compression in the \(x\) -direction with factor \(\frac{1}{3}\). b. A reflection about the line \(y=x\) and an expansion in the \(x\) -direction with factor \(2\). Equation Transcription: Text Transcription: x-axis x 1 \ 3 y=x x 2

In each part of Exercises 9-10, determine whether the stated operators commute. a. A shear in the \(y\) -direction by a factor \(\frac{1}{4}\) and a shear in the \(y\) -direction by a factor \(\frac{3}{5}\) b. A shear in the \(y\)-direction by a factor \(\frac{1}{4}\) and a shear in the \(x\) -direction by a factor \(\frac{3}{5}\) Equation Transcription: Text Transcription: y 1 \ 4 y 3 \ 5 y 1 \ 4 x 3 \ 5

In Exercises 11-14, express the matrix as a product of elementary matrices, and then describe the effect of multiplication by A in terms of shears, compressions, expansions, and reflections. \(A=\left[\begin{array}{cc} 4 & 4 \\ 0 & -2 \end{array}\right]\) Equation Transcription: [] Text Transcription: A=[ 4 4 \ 0 -2 ]

In Exercises 11-14, express the matrix as a product of elementary matrices, and then describe the effect of multiplication by A in terms of shears, compressions, expansions, and reflections. \(A=\left[\begin{array}{ll} 1 & 4 \\ 2 & 9 \end{array}\right]\) Equation Transcription: [] Text Transcription: A=[ 1 4 \ 2 9 ]

In Exercises 11-14, express the matrix as a product of elementary matrices, and then describe the effect of multiplication by A in terms of shears, compressions, expansions, and reflections. \(A=\left[\begin{array}{rr} 0 & -2 \\ 4 & 0 \end{array}\right]\) Equation Transcription: [] Text Transcription: A=[ 0 -2 \ 4 0 ]

In Exercises 11-14, express the matrix as a product of elementary matrices, and then describe the effect of multiplication by A in terms of shears, compressions, expansions, and reflections. \(A=\left[\begin{array}{lr} 1 & -3 \\ 4 & 6 \end{array}\right]\) Equation Transcription: [] Text Transcription: A=[ 1 -3 \ 4 6 ]

In each part of Exercises 15–16, describe, in words, the effect on the unit square of multiplication by the given diagonal matrix. (a) \(A=\left[\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right]\) (b) \(A=\left[\begin{array}{cc} 1 & 0 \\ 0 & -5 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: A=[ 3 0 \ 0 1 ] A=[ 1 0 \ 0 -5 ]

In each part of Exercises 15–16, describe, in words, the effect on the unit square of multiplication by the given diagonal matrix. (a) \(A=\left[\begin{array}{rr} -2 & 0 \\ 0 & 1 \end{array}\right]\) (b) \(A=\left[\begin{array}{rr} -3 & 0 \\ 0 & -1 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: A=[ -2 0 \ 0 1 ] A=[ -3 0 \ 0 -1 ]

a. Show that multiplication by \(A=\left[\begin{array}{ll}3 & 1 \\6 & 2\end{array}\right]\) maps each point in the plane onto the line \(y=2 x\) b. It follows from part (a) that the noncollinear points (1,0), (0,1),(-1,0) are mapped onto a line. Does this violate part ( \(e\) ) of Theorem 8.6.1? Equation Transcription: Text Transcription: A = [3 & 1 \\6 & 2] y = 2x e

Find the matrix for a shear in the \(x\)-direction that transforms the triangle with vertices (0, 0), (2, 1), and (3, 0) into a right triangle with the right angle at the origin. Equation Transcription: Text Transcription: x

In accordance with part (\(c\)) of Theorem 8.6.1, show that multiplication by the invertible matrix \(A=\left[\begin{array}{ll}3 & 2 \\1 & 1\end{array}\right]\) maps the parallel lines \(y=3 x+1\) and \(y=3 x-2\) into parallel lines. Equation Transcription: Text Transcription: c A = [3 & 2 \\1 & 1] y = 3x + 1 y = 3x - 2

Draw a figure that shows the image of the triangle with vertices (0, 0), (1, 0), and (0.5, 1) under a shear by a factor of 2 in the \(x\)-direction. Equation Transcription: Text Transcription: x

a. Draw a figure that shows the image of the triangle with vertices \((0,0),(1,0)\), and \((0.5,1)\) under multiplication by \(A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right]\) b. Find a succession of shears, compressions, expansions, and reflections that produces the same image. Equation Transcription: Text Transcription: (0,0),(1,0) (0.5,1) A=[1 & 1\\1 & -1]

Find the endpoints of the line segment that results when the line segment from \(P(1,2) \) to \(Q(3,4)\) is transformed by a. a compression with factor \(\frac{1}{2}\) in the \(y\) -direction. b. a rotation of \(30^{\circ}\) about the origin. Equation Transcription: Text Transcription: P(1,2) Q(3,4) 1/2 y 30 degree

Draw a figure showing the italicized letter " that results when the letter in the accompanying figure is sheared by a factor \(\frac{1}{4}\) in the \(x\)-direction. Equation Transcription: Text Transcription: 1/4 x

Can an invertible matrix operator on \(R^{2}\) map a square region into a triangular region? Justify your answer. Equation Transcription: Text Transcription: R^2

Find the image of the triangle with vertices \((0,0),(1,1),(2,0)\) under multiplication by \(A=\left[\begin{array}{rr} 2 & -1 \\ 0 & 0 \end{array}\right]\) Does your answer violate part of Theorem ? Explain. Equation Transcription: Text Transcription: (0,0),(1,1),(2,0) A=[0 & 0\\2 & -1]

In \(R^{3}\) the shear in the xy-direction by a factor \(k\) is the matrix transformation that moves each point \((x,y,z)\) parallel to the \(xy\) -plane to the new position \((x+kz,y+kz,z)\). (See the accompanying figure.) a. Find the standard matrix for the shear in the \(xy\) -direction by a factor \(k\). b. How would you define the shear in the \(xz\)-direction by a factor \(k\) and the shear in the \(yz\)-direction by a factor \(k\)? What are the standard matrices for these matrix transformations? Equation Transcription: Text Transcription: (x,y,z) R^3 k xy (x+kz,y+kz,z)

In Exercises 27-28, find the standard matrix for the operator \(T: R^{3} \rightarrow R^{3}\) that performs the stated rotation. a. rotates each vector \(30^{\circ}\) counterclockwise about the \(z\)-axis (looking along the positive \(z\)-axis toward the origin). b. rotates each vector \(45^{\circ}\) clockwise about the \(x\)-axis (looking along the positive \(x\)-axis toward the origin). Equation Transcription: Text Transcription: T:R^3 -> R^3 30 degree 45 degree x z

In Exercises 27-28, find the standard matrix for the operator \(T: R^{3} \rightarrow R^{3}\) that performs the stated rotation. a. rotates each vector \(90^{\circ}\) counterclockwise about the \(y\)-axis (looking along the positive \(y\) -axis toward the origin). b. rotates each vector \(90^{\circ}\) clockwise about the positive \(z\)-axis looking toward the origin. Equation Transcription: Text Transcription: T:R^3 -> R^3 y z 90 degree

Use Formula (3) to find the standard matrix for a rotation of \(180^{\circ}\) about the axis determined by the vector \(v=(2,2,1)\) [Note: Formula (3) requires that the vector defining the axis of rotation have length Equation Transcription: Text Transcription: 180 degree v=(2,2,1)

Use Formula (3) to find the standard matrix for a rotation of \(\pi / 2\) radians about the axis determined by \(v=(1,1,1)\). [Note: Formula (3) requires that the vector defining the axis of rotation have length 1.] Equation Transcription: Text Transcription: pi/2 v=(1,1,1)

Use Formula (3) to derive the standard matrices for the rotations about the \(x\)-axis, the \(y\) -axis, and the \(z\)-axis through an angle of \(90^{\circ}\) in \(R^{3}\). Equation Transcription: Text Transcription: R^3 x y z 90 degree

Euler's Axis of Rotation Theorem states: If \(A\) is an orthogonal \(3 \times 3\) matrix for which \(det?(A)=1\), then multiplication by \(A\) is a rotation about a line through the origin in \(R^{3}\). Moreover, if \(u\) is a unit vector along this line, then \(Au=u\) a. Confirm that the following matrix \(A\) is orthogonal, that \(det?(A)=1\), and that there is a unit vector \(u\) for which \(Au=u\) \(A=\left[\begin{array}{rrr} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\ \frac{3}{7} & -\frac{6}{7} & \frac{2}{7} \\ \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \end{array}\right]\) b. Use Formula (3) to prove that if \(A\) is a \(3 \times 3\) orthogonal matrix for which \(det?(A)=1\), then the angle of rotation resulting from multiplication by \(A\) satisfies the equation \(\cos \theta=\frac{1}{2}[\operatorname{tr}(A)-1\)]. Use this result to find the angle of rotation for the rotation matrix in part (a). Equation Transcription: Text Transcription: A 3×3 det?(A)=1 R^3 u Au=u Cos? theta=1/2[tr?(A)-1] A=[2/7 & 3/7 & 6/7\\ 3/7 & -6/7 & 2/7\\ 6/7 & 2/7 & -3/7]

Prove part of Theorem 8.6.1. [Hint: A line in the plane has an equation of the form \(Ax+By+C=0\), where \(A\) and \(B\) are not both zero. Use the method of example 1 to show that the image of this line under multiplication by the invertible matrix \(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) has the equation \(A' x+B' y+C=0\), where \(A^{\prime}=(d A-c B) /(a d-b c)\) and \(B^{\prime}=(-b A+a B) /(a d-b c)\) Then show that \(A^{\prime}\) and \(B^{\prime}\) are not both zero to conclude that the image is a line. Equation Transcription: Text Transcription: Ax+By+C=0 A B A'x+B'y+C=0 A'=(dA-cB)/(ad-bc) B'=(-bA+aB)/(ad-bc) A' B' [c & d\\a & b]

Use the hint in Exercise 33 to prove parts and of Theorem 8.6.1. Equation Transcription: Text Transcription:

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. The image of the unit square under a one-to-one matrix operator is a square.

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. A \(2 \times 2\) invertible matrix operator has the geometric effect of a succession of shears, compressions, expansions, and reflections. Equation Transcription: Text Transcription: 2 times 2

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. The image of a line under an invertible matrix operator is a line.

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. Every reflection operator on \(R^{2}\) is its own inverse. Equation Transcription: Text Transcription: R^2

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. The matrix \(\left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right]\) represents reflection about a line. Equation Transcription: Text Transcription: [1 & -1\\1 & 1]

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. The matrix \(\left[\begin{array}{rr} 1 & -2 \\ 2 & 1 \end{array}\right]\) represents a shear. Equation Transcription: Text Transcription: [1 & -2 \\ 2 & 1]

In parts (a)-(g) determine whether the statement is true or false, and justify your answer. The matrix \(\left[\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right]\) represents an expansion. Equation Transcription: Text Transcription: [1 & 0 \\ 0 & 3]

a. Find the standard matrix for the linear operator on \(R^{3}\) that performs a counterclockwise rotation of \(47^{\circ}\) about the \(x\)-axis, followed by a counterclockwise rotation of \(68^{\circ}\) about the \(y\)-axis, followed by a counterclockwise rotation of \(33^{\circ}\) about the \(z\) -axis. b. Find the image of the point \((1,1,1)\) under the operator in part (a). Equation Transcription: Text Transcription: R^3 47 degree x 68 degree y 33 degree (1,1,1) z

Find the standard matrix for the linear operator on \(R^{2}\) that first reflects each point in the plane about the line through the origin that makes an angle of \(27^{\circ}\) with the positive \(x\)--axis and then projects the resulting point orthogonally onto the line through the origin that makes an angle of \(51^{\circ}\) with the positive \(x\)-axis. Equation Transcription: Text Transcription: R^2 27 degree x 51 degree