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Chapter 6: Problem 17

Find all fixed points of the linear transformation. Recall that the vector \(\mathbf{v}\) is a fixed pointof \(T\) when \(T(v)=v\). A reflection in the line \(y=x\)

Short Answer

Expert verified
All points on the line \(y=x\) are fixed points of the linear transformation.

Step by step solution

01

Identify the Transformation Matrix

The reflection in the line \(y=x\) can be represented as a 2x2 transformation matrix \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \). This matrix is derived from the transformation \(T(x, y) = (y, x)\) which happens when we reflect any point in the line \(y=x\).
02

Determine the Fixed Points

To find the fixed points v of the transformation T, we need to solve the equation \(T(v) = v\) or equivalently \(Av = v\) where \(v = (x, y)\) is a point in 2D space. This gives us the system of equations: \(0 * x + 1 * y = x\) and \(1 * x + 0 * y = y\). Simplifying, we get \(y = x\) and \(x = y\), both of which are satisfied by any point on the line \(y = x\).
03

State the Solution

Since any point on the line \(y = x\) satisfies the equation \(y = x\) and \(x = y\), it follows that every point on the line \(y = x\) is a fixed point of the transformation.

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Key Concepts

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

Understanding Reflection
Reflection is a type of linear transformation that flips points over a specified axis or line. In our case, the line of reflection is defined as the equation \(y = x\). When you reflect a point across this line, its x-coordinate and y-coordinate are swapped.
A reflection transformation has the following properties:
  • Symmetry: Points equidistant from the line \(y = x\) will map to each other.
  • Reversal: The orientation of shapes is reversed after reflection.
  • Invariance of the line: Points lying on the line \(y = x\) remain unchanged.
By understanding these properties, we can better grasp how individual points and larger shapes are transformed by reflection operations.
Exploring Fixed Points
A fixed point of a transformation is a point that remains unchanged even after undergoing the transformation. In mathematical terms, if \(T(v) = v\) where \(T\) is the transformation and \(v\) is the point, then \(v\) is a fixed point.
For the given transformation, which is a reflection across the line \(y = x\), we find fixed points by solving \(Av = v\) where \(A\) is the transformation matrix.
  • The equation simplifies to \(y = x\), indicating that all points on this line act as their own mirrors, thus do not move.
  • The whole line \(y = x\) consists of fixed points as each point maintains its position.
  • This reflects the intuitive notion that a point on the line of reflection does not change during the reflection.
Identifying fixed points is important as it helps in understanding how transformations behave in 2-dimensional space.
Decoding Transformation Matrix
The transformation matrix is a mathematical tool that represents linear transformations including reflections, rotations, and scaling. For this exercise, we use the transformation matrix:\[A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]as it represents reflection over the line \(y = x\).
Here's how it works:
  • When you multiply this matrix by a vector \((x, y)\), the result is \((y, x)\), effectively swapping the coordinates.
  • This matrix, thus, provides a succinct way of expressing the reflection process.
  • Understanding transformation matrices is crucial as they form the basis for manipulating geometric figures in linear algebra.
By mastering how to use these matrices, one can efficiently perform and analyze various linear transformations.

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Most popular questions from this chapter

Let \(V\) be an inner product space with a subspace \(W\) having \(B=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{n}\right\\}\) as an orthonormal basis. Show that the function \(T: V \rightarrow W\) represented by \(T(\mathbf{v})=\left\langle\mathbf{v}, \mathbf{w}_{1}\right\rangle \mathbf{w}_{1}+\left\langle\mathbf{v}, \mathbf{w}_{2}\right\rangle \mathbf{w}_{2}+\cdots+\left\langle\mathbf{v}, \mathbf{w}_{n}\right\rangle \mathbf{w}_{n}\) is a linear transformation. \(T\) is called the orthogonal projection of \(V\) onto \(W\)

Sketch the image of the rectangle with vertices at \((0,0),(1,0),(1,2),\) and \((0,2)\) under the specified transformation. \(T\) is the expansion represented by \(T(x, y)=(x, 6 y)\)

Give a geometric description of the linear transformation defined by the elementary matrix. $$A=\left[\begin{array}{ll}1 & 0 \\\0 & \frac{1}{2}\end{array}\right]$$

Find the image of the vector \((1,1,1)\) for the rotation. \(60^{\circ}\) about the \(y\) -axis

find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) $$ \begin{aligned} &T: R^{3} \rightarrow R^{3}, T(x, y, z)=(0,0,0)\\\ &B^{\prime}=\\{(1,1,0),(1,0,1),(0,1,1)\\} \end{aligned} $$
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