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Chapter 6: Problem 15

Finding the Standard Matrix and the Image In Exercises \(11-22,\) (a) find the standard matrix \(A\) for the linear transformation \(T,\) (b) use \(A\) to find the image of the vector \(\mathbf{v},\) and (c) sketch the graph of \(\mathbf{v}\) and its image. \(T\) is the counterclockwise rotation of \(45^{\circ}\) in \(R^{2}\) \(\mathbf{v}=(2,2)\)

Short Answer

Expert verified
The standard matrix A for the counterclockwise rotation of \( 45^{\circ} \) in \(R^{2}\) is \( \left(\begin{array}{cc} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}\right) \). The image of the vector v=(2,2) is (0, \(2\sqrt{2}\)) after a counterclockwise rotation by \(45掳\).

Step by step solution

01

Finding Standard Matrix A

For a linear transformation rotating counterclockwise by 胃 degrees in 鈩澛�, the standard matrix A is structured as follows - \[ A= \left(\begin{array}{cc} \cos(胃) & -\sin(胃) \\ \sin(胃) & \cos(胃) \end{array}\right) \] Plug in the value of \(胃=45掳=\frac{蟺}{4} rad\) to the above matrix. We have, \[ A= \left(\begin{array}{cc} \cos(\frac{蟺}{4}) & -\sin(\frac{蟺}{4}) \\ \sin(\frac{蟺}{4}) & \cos(\frac{蟺}{4}) \end{array}\right) \] Simplifying the trigonometric values we get, \[ A= \left(\begin{array}{cc} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}\right) \] So, the standard matrix \(A\) for the counterclockwise rotation of \( 45^{\circ} \) in \(R^{2}\) is \( \left(\begin{array}{cc} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}\right) \)
02

Find the image of the vector using A

The image of the vector \(\mathbf{v}\) under the transformation \(T\) is given by multiplying the standard matrix \(A\) with the vector \(\mathbf{v}\). The multiplication is as follows: \[ \begin{array}{cc} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}\right) \times \left(\begin{array}{c} 2 \\ 2 \end{array}\right)=\left(\begin{array}{c} 0 \\ 2\sqrt{2} \end{array}\right) \] So, the image of the vector \(\mathbf{v}\)=(2,2) after rotation by \(45掳\) counterclockwise is (0, \(2\sqrt{2}\)).
03

Sketching the vectors and their transformation

To sketch the vectors, put the tip of vector \(\mathbf{v}\) at point (2,2) and vector T(\(\mathbf{v}\)) or the image of \(\mathbf{v}\) under transformation T at point (0, \(2\sqrt{2}\)). Both the vectors will originate from the same point and the image vector will be rotated \(45掳\) counterclockwise from the original vector \(\mathbf{v}\).

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Most popular questions from this chapter

use the matrix \(P\) to show that the matrices \(A\) and \(A^{\prime}\) are similar. $$ P=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right], A=\left[\begin{array}{lll} 5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{array}\right], A^{\prime}=\left[\begin{array}{lll} 5 & 2 & 2 \\ 0 & 3 & 2 \\ 0 & 0 & 1 \end{array}\right] $$

Determine whether the function involving the \(n \times n\) matrix \(A\) is a linear transformation. $$T: M_{n, n} \rightarrow R, T(A)=a_{11} \cdot a_{22} \cdot \cdots \cdot a_{n n}, \text { where } A=\left[a_{i j}\right]$$

find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) $$ \begin{aligned} &T: R^{2} \rightarrow R^{2}, T(x, y)=(5 x+4 y, 4 x+5 y)\\\ &B^{\prime}=\\{(12,-13),(13,-12)\\} \end{aligned} $$

A translation in \(R^{2}\) is a function of the form \(T(x, y)=(x-h, y-k),\) where at least one of the constants \(h\) and \(k\) is nonzero. (a) Show that a translation in \(R^{2}\) is not a linear transformation. (b) For the translation \(T(x, y)=(x-2, y+1)\) determine the images of \((0,0),(2,-1),\) and \((5,4)\) (c) Show that a translation in \(R^{2}\) has no fixed points.

Prove that \(T: V \rightarrow W\) is a linear transformation if and only if \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\) for all vectors \(\mathbf{u}\) and \(\mathbf{v}\) and all scalars \(a\) and \(b\) Getting Started: This is an "if and only if" statement, so you need to prove the statement in both directions. To prove that \(T\) is a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let \(T\) be a linear transformation. Use the definition and properties of a linear transformation to prove that \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\). (i) Let \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v}),\) Show that \(T\) preserves the properties of vector addition and scalar multiplication by choosing appropriate values of \(a\) and \(b\). (ii) To prove the statement in the other direction, assume that \(T\) is a linear transformation. Use the properties and definition of a linear transformation to show that \(T(a \mathbf{u}+b \mathbf{v})=a T(\mathbf{u})+b T(\mathbf{v})\).
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