/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Let \(T: R^{2} \rightarrow R^{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 2

Let \(T: R^{2} \rightarrow R^{2}\) be a reflection in the \(y\) -axis. Find the image of each vector. (a) (2,5) (b) (-4,-1) (c) \((a, 0)\) (d) \((0, b)\) (e) \((c,-d)\) (f) \((f, g)\)

Short Answer

Expert verified
The images of the vectors under transformation \(T\) reflection over the y-axis are as follows: (a) \((-2,5)\), (b) \((4,-1)\), (c) \((-a,0)\), (d) \((0,b)\), (e) \((-c,-d)\), and (f) \((-f,g)\).

Step by step solution

01

Reflect the Vector (2,5)

The image of the vector (2,5), under the transformation \(T\), becomes \((-2,5)\). This is because only the 'x' component of the vector changes its sign in reflection over the y-axis.
02

Reflect the Vector (-4,-1)

Following the same logic, the image of the vector (-4,-1) under the transformation \(T\) becomes \((4,-1)\).
03

Reflect the general Vector (a,0)

For a general vector like (a,0), the image through \(T\) becomes \((-a,0)\). As the vector lies on the x-axis, the reflected vector will also lie on the x-axis but in the opposite direction.
04

Reflect the Vector (0,b)

Now, when the vector is (0,b), which lies on y-axis, the reflection doesn't change its position. Thus, the image of (0,b) under \(T\) remains (0,b).
05

Reflect the general Vector (c,-d)

When it comes to the general vector (c,-d), its reflection under the transformation \(T\) will result in \((-c,-d)\).
06

Reflect the most general Vector (f,g)

For the most general case, when the vector is (f,g), its transformation through \(T\) will produce \((-f,g)\). As always, only the 'x' component of the vector changes its sign while reflecting over the y-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vector Reflection
Vector reflection is a type of geometric transformation where each point of a vector is mapped to a new location over a line, known as the axis of reflection. In simpler terms, imagine placing a mirror along a certain line; the reflection you see of a vector in this mirror is the new vector in this transformation.
The exercise involves reflecting vectors over the y-axis in a plane. This means the x-component of any given vector is negated, while the y-component remains unchanged. Here are the notable points to remember for reflection in the y-axis:
  • The transformation affects only the x-component of a vector.
  • The y-component of the vector stays the same during reflection.
  • For example, a vector \( (a, b) \) would transform into \((-a, b)\) when reflected over the y-axis.
  • This type of transformation is a linear transformation, as it alters the orientation of vectors in a linear manner, without affecting distances or angles between vectors.
Understanding this concept is crucial as it forms the basis for more complex geometrical transformations in multidimensional spaces.
Coordinate Transformation
Coordinate transformations involve changing the position of points or vectors in a coordinate system to reflect a given transformation or displacement. They are fundamental in geometry, physics, and engineering wherever spatial relationships are crucial.
In our exercise, the coordinate transformation is realized by reflecting vectors over the y-axis. This specific transformation alters only one coordinate:
  • The x-component (horizontal coordinate) of each point is flipped to its negative counterpart.
  • The y-component (vertical coordinate) remains unchanged, maintaining the vector's vertical placement.
  • Such transformations simplify complex spatial computations by reorienting vectors in a consistent and predictable manner.
  • This is best visualized as folding a piece of paper along the y-axis and placing each vector's image directly onto the opposite side of the axis.
Coordinate transformations are key to understanding how objects can be repositioned or reoriented in space through mathematical manipulations.
Geometric Transformations
Geometric transformations involve moving, scaling, or rotating figures in a plane. These transformations are used in various fields such as computer graphics, robotics, and physics to model and simulate motion and change in real-world systems.
Reflection is a fundamental type of geometric transformation. It creates a mirror image of the original object across a specified line, known in this case as the y-axis. Key aspects of geometric transformations, particularly vector reflections, include:
  • Preservation of shape and size: The reflected vector retains its original length and form.
  • Change in orientation: The reflection alters the object's orientation relative to the axis.
  • Application in symmetry: Reflections are crucial for creating symmetrical patterns and designs.
  • Transformations like reflection are linear, meaning they preserve lines and parallelism.
Mastering geometric transformations enables one to understand and predict how objects interact within a given space, aiding in problem-solving and design tasks across multiple disciplines.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) and \((b)\) show that \(A^{\prime}\) is similar to \(A,\) the standard matrix for \(T\). $$\begin{array}{l} T: R^{3} \rightarrow R^{3}, T(x, y, z)=(x-y+2 z, 2 x+y-z, x+2 y+z) \\ B^{\prime}=\\{(1,0,1),(0,2,2),(1,2,0)\\} \end{array}$$

Find the matrix that will produce the indicated rotation. \(60^{\circ}\) about the \(x\) -axis

Find \(T(\mathbf{v})\) by using (a) the standard matrix and (b) the matrix relative to \(B\) and \(B^{\prime}\). $$\begin{array}{l} T: R^{3} \rightarrow R^{2}, T(x, y, z)=(x-y, y-z), \mathbf{v}=(1,2,-3) \\ B=\\{(1,1,1),(1,1,0),(0,1,1)\\}, B^{\prime}=\\{(1,2),(1,1)\\} \end{array}$$

Let \(A\) be an \(n \times n\) matrix such that \(A^{2}=O .\) Prove that if \(B\) is similar to \(A,\) then \(B^{2}=O\).

Find all fixed points of the linear transformation. The vector \(\mathbf{v}\) is a fixed point of \(T\) if \(T(\mathbf{v})=\mathbf{v}\). A reflection in the \(x\) -axis
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.