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Chapter 1: Problem 9

In Exercises \(1-10\) , assume that \(T\) is a linear transformation. Find the standard matrix of \(T\) . \(T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) first performs a horizontal shear that transforms \(\mathbf{e}_{2}\) into \(\mathbf{e}_{2}-2 \mathbf{e}_{1}\) (leaving \(\mathbf{e}_{1}\) unchanged) and then reflects points through the line \(x_{2}=-x_{1}\)

Short Answer

Expert verified
The standard matrix is \( \begin{pmatrix} 2 & -1 \\ -1 & 0 \end{pmatrix} \).

Step by step solution

01

Define the matrix for horizontal shear

To perform a horizontal shear that transforms \( \mathbf{e}_{2}\) into \( \mathbf{e}_{2} - 2 \mathbf{e}_{1} \), we use the shear matrix \( A \). This leaves \( \mathbf{e}_{1} \) unchanged, therefore its matrix representation is given by:\[ A = \begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix} \]
02

Define the matrix for reflection through line

Reflecting points through the line \( x_{2} = -x_{1} \) can be represented by the matrix \( B \):\[ B = \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix} \]
03

Calculate the standard matrix of T

To find the standard matrix of the linear transformation \( T \), which involves the shear followed by the reflection, we need to multiply the matrices \( B \) and \( A \) in that order:\[ C = B \times A = \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix} = \begin{pmatrix} 2 & -1 \ -1 & 0 \end{pmatrix} \]
04

Write the final answer

The standard matrix \( C \) of the transformation \( T \) is:\[ \begin{pmatrix} 2 & -1 \ -1 & 0 \end{pmatrix} \]

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Key Concepts

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

Understanding Horizontal Shear
A horizontal shear in a linear transformation is a subtle yet fascinating effect. Imagine pushing an object sideways, while its height stays the same. This is essentially what horizontal shear does. It transforms points in a way that alters their horizontal position but keeps vertical ones untouched.
For instance, when applying a horizontal shear matrix on the vector basis
  • \( \mathbf{e}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix} \): This vector remains unchanged during the shear.
  • \( \mathbf{e}_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix} \): This vector changes to \( \begin{pmatrix} -2 \ 1 \end{pmatrix} \), indicating that an additional force of \(-2\mathbf{e}_1\) is added in the horizontal direction.
The standard matrix for this horizontal shear can be given as:\[A = \begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix}\] This no-frills matrix visually captures the essence of horizontal shear transformations in linear algebra.
Reflection Matrix Simplified
When we reflect across a line, we are essentially flipping points in a defined manner. It is like seeing your image in a mirror, but the reflection doesn鈥檛 always follow common intuition. Here, the line of reflection is \( x_2 = -x_1 \), a diagonal line chasing down the quadrants of the Cartesian plane.
A reflection matrix that handles this transformation needs to navigate this diagonal line efficiently. It is represented as:\[B = \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}\]
  • This matrix demonstrates how each point \( (x_1, x_2) \) is re-oriented to \( (-x_2, -x_1) \).
  • As you can see, the matrix entries mandate a swap and flip between axes, ensuring points mirror accurately over \( x_2 = -x_1 \).
Geometry lovers can relate to this flip-and-swap action, a critical part of working with reflection matrices.
Navigating the Standard Matrix
In any linear transformation, understanding the standard matrix is vital. It鈥檚 your map that gives a clear picture of how the transformation behaves across vectors.
For transformations involving a sequence of other operations鈥攍ike shear and reflection in our example鈥攜ou check out the ultimate impact through multiplication of individual matrices.
  • Start with your reflection matrix \( B \), then multiply the shear matrix \( A \):\[C = B \times A = \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix} = \begin{pmatrix} 2 & -1 \ -1 & 0 \end{pmatrix}\]
  • Here, \( C \) is the standard matrix of transformation \( T \).
  • This concise matrix portrays the cumulative effects鈥攆irst transforming via shear, followed by reflection.
This approach underscores the practicality of matrix multiplication in visualizing and executing linear transformations seamlessly.

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Most popular questions from this chapter

In a certain region, about 7\(\%\) of a city's population moves to the surrounding suburbs each year, and about 5\(\%\) of the suburban population moves into the city. In 2015 , there were \(800,000\) residents in the city and \(500,000\) in the suburbs. Set up a difference equation that describes this situation, where \(\mathbf{x}_{0}\) is the initial population in 2015 . Then estimate the populations in the city and in the suburbs two years later, in 2017 . (Ignore other factors that might influence the population sizes.)

Construct a \(3 \times 3\) matrix, not in echelon form, whose columns do not span \(\mathbb{R}^{3}\) . Show that the matrix you construct has the desired property.

Exercises \(17-20\) refer to the matrices \(A\) and \(B\) below. Make appropriate calculations that justify your answers and mention an appropriate theorem. $$ A=\left[\begin{array}{rrrr}{1} & {3} & {0} & {3} \\ {-1} & {-1} & {-1} & {1} \\\ {0} & {-4} & {2} & {-8} \\ {2} & {0} & {3} & {-1}\end{array}\right] \quad B=\left[\begin{array}{rrrr}{1} & {3} & {-2} & {2} \\ {0} & {1} & {1} & {-5} \\\ {1} & {2} & {-3} & {7} \\ {-2} & {-8} & {2} & {-1}\end{array}\right] $$ How many rows of \(A\) contain a pivot position? Does the equation \(A \mathbf{x}=\mathbf{b}\) have a solution for each \(\mathbf{b}\) in \(\mathbb{R}^{4} ?\)

[M] In Exercises \(37-40\) , determine if the columns of the matrix span \(\mathbb{R}^{4} .\) $$ \left[\begin{array}{rrrr}{7} & {2} & {-5} & {8} \\ {-5} & {-3} & {4} & {-9} \\\ {6} & {10} & {-2} & {7} \\ {-7} & {9} & {2} & {15}\end{array}\right] $$

The given matrix determines a linear transformation \(T .\) Find all \(\mathbf{x}\) such that \(T(\mathbf{x})=\mathbf{0} .\) \(\left[\begin{array}{rrrr}{4} & {-2} & {5} & {-5} \\ {-9} & {7} & {-8} & {0} \\\ {-6} & {4} & {5} & {3} \\ {5} & {-3} & {8} & {-4}\end{array}\right]\)
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