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Chapter 4: Problem 7

Find the standard matrix for the stated composition in \(R^{3}\). (a) A reflection about the \(y z\) -plane, followed by an orthogonal projection on the \(x z\) -plane. (b) A rotation of \(45^{\circ}\) about the \(y\) -axis, followed by a dilation with factor \(k=\sqrt{2}\) (c) An orthogonal projection on the \(x y\) -plane, followed by a reflection about the \(y z\) -plane.

Short Answer

Expert verified
(a) \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \), (b) \( \begin{bmatrix} 1 & 0 & 1 \\ 0 & \sqrt{2} & 0 \\ -1 & 0 & 1 \end{bmatrix} \), (c) \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)

Step by step solution

01

Identify Reflection Matrix over yz-plane

Reflection over the \( yz \)-plane in \( R^3 \) modifies the sign of the corresponding \( x \)-coordinate. Thus, the reflection matrix is:\[R_{yz} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
02

Identify Orthogonal Projection Matrix on xz-plane

For projecting onto the \( xz \)-plane, the \( y \)-coordinate is zeroed. The projection matrix is:\[P_{xz} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
03

Combine Transformations for (a)

The standard matrix for the composition of reflections followed by projections is their product. First, apply the reflection, then the projection:\[A_1 = P_{xz} \cdot R_{yz} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
04

Identify Rotation Matrix about y-axis

A rotation about the \( y \)-axis by \( 45^{\circ} \) is given by the following matrix:\[R_y(45^{\circ}) = \begin{bmatrix} \cos 45^{\circ} & 0 & \sin 45^{\circ} \ 0 & 1 & 0 \ -\sin 45^{\circ} & 0 & \cos 45^{\circ} \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \ 0 & 1 & 0 \ -\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{bmatrix}\]
05

Identify Dilation Matrix with Factor k

The dilation matrix with factor \( k = \sqrt{2} \) for \( R^3 \) is a scalar multiplication of identity:\[D = \sqrt{2} \cdot I = \begin{bmatrix} \sqrt{2} & 0 & 0 \ 0 & \sqrt{2} & 0 \ 0 & 0 & \sqrt{2} \end{bmatrix}\]
06

Combine Transformations for (b)

The standard matrix for the composition involves first rotating, followed by dilation:\[A_2 = D \cdot R_y(45^{\circ}) = \begin{bmatrix} \sqrt{2} & 0 & 0 \ 0 & \sqrt{2} & 0 \ 0 & 0 & \sqrt{2} \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \ 0 & 1 & 0 \ -\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \ 0 & \sqrt{2} & 0 \ -1 & 0 & 1 \end{bmatrix}\]
07

Identify Orthogonal Projection Matrix on xy-plane

To project on the \( xy \)-plane, zero the \( z \)-coordinate with the following matrix:\[P_{xy} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}\]
08

Combine Transformations for (c)

First apply the projection onto the \( xy \)-plane, then reflect over the \( yz \)-plane:\[A_3 = R_{yz} \cdot P_{xy} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection About Planes
In geometry, reflection involves flipping an object over a line or a plane, creating a mirror image. Imagine folding a piece of paper along a line; one side of the paper reflects onto the other. In linear transformations, reflection about a plane is represented using a specific matrix. Let's look at the reflection about the yz-plane in three-dimensional space (R^3).
  • The yz-plane reflection changes the sign of the x-coordinate, keeping the y and z coordinates the same.
  • This transformation does not affect values along the y or z-axis, maintaining their coordinates.
The standard matrix for the reflection about the yz-plane is:\[R_{yz} = \begin{bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]This matrix effectively flips the x-component of any point across the yz-plane.
Orthogonal Projection
Orthogonal projection is a type of linear transformation used to reduce the dimensions of a vector by dropping one or more of its components. It projects a vector onto a subspace like a plane or a line.
  • For example, projecting onto the xz-plane involves eliminating the y-component of a vector.
  • The orthogonal projection matrix onto this plane is characterized by having zeros in place of the y-component.
The standard matrix employed for projecting onto the xz-plane is given by:\[P_{xz} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}\]This transformation keeps the x and z coordinates unchanged while setting the y-coordinate to zero, effectively mapping the vector onto the xz-plane.
Rotation Matrix
Rotation matrices are essential tools in linear algebra, allowing us to rotate a vector around an axis in space. When rotating a vector within R^3, it's often done about one of the three main axes (x, y, or z).
  • In this context, a 45-degree rotation about the y-axis changes the x and z components based on trigonometric functions sine and cosine.
  • This rotation matrix is constructed using those trigonometric components to ensure proper orientation post-rotation.
The rotation matrix for a 45-degree rotation about the y-axis is represented as:\[R_y(45^{\circ}) = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \ 0 & 1 & 0 \ -\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{bmatrix}\]It reorients the vector by mixing its x and z components following the rules of 2D rotation within the xz-plane.
Dilation in Linear Transformations
Dilation is a transformation that changes the size of a figure but maintains its shape and orientation. In linear transformations, scaling or dilation can be applied using a scalar factor across all dimensions.
  • A dilation transformation enlarges or shrinks a vector by a specific factor without altering its direction.
  • In R^3, dilation applies universally across x, y, and z axes using an identity matrix scaled by this factor.
The matrix for dilation with factor \( k = \sqrt{2} \) is:\[D = \begin{bmatrix} \sqrt{2} & 0 & 0 \ 0 & \sqrt{2} & 0 \ 0 & 0 & \sqrt{2} \end{bmatrix}\]With this transformation, each component of a vector is scaled by \( \sqrt{2} \), thus dilating its overall magnitude.

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Most popular questions from this chapter

In words, describe the geometric effect of multiplying a vector \(x\) by the matrix $$A=\left[\begin{array}{cc} \cos ^{2} \theta-\sin ^{2} \theta & -2 \sin \theta \cos \theta \\ 2 \sin \theta \cos \theta & \cos ^{2} \theta-\sin ^{2} \theta \end{array}\right]$$

Construct a matrix whose null space consists of all linear combinations of the vectors $$\mathbf{v}_{1}=\left[\begin{array}{r} 1 \\ -1 \\ 3 \\ 2 \end{array}\right] \text { and } \mathbf{v}_{2}=\left[\begin{array}{r} 2 \\ 0 \\ -2 \\ 4 \end{array}\right]$$

\- Find the standard matrix for the given matrix operator. (a) \(T: R^{3} \rightarrow R^{3}\) reflects a vector about the \(x z\) -plane and then contracts that vector by a factor of \(\frac{1}{5}\) (b) \(T: R^{3} \rightarrow R^{3}\) projects a vector orthogonally onto the \(x z\) -plane and then projects that vector orthogonally onto the \(x y\) -plane. (c) \(T: R^{3} \rightarrow R^{3}\) reflects a vector about the \(x y\) -plane, then reflects that vector about the \(x z\) -plane, and then reflects that vector about the \(y z\) -plane.

Use matrix multiplication to find the reflection of (2,-5,3) about (a) the \(x y\) -plane. (b) the \(x z\) -plane. (c) the \(y z\) -plane.

Use matrix multiplication to find the reflection of (-1,2) about (a) the \(x\) -axis. (b) the \(y\) -axis. (c) the line \(y=x\).
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