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Rotations and reflections have two remarkable properties: They preserve the length of vectors and the angle between vectors. (Draw figures illustrating these properties.) We will show that, conversely, any linear transformationT from ${}^{\mathbf{2}}$to${}^{\mathbf{2}}$ that preserves length and angles is either a rotation or a reflection (about a line).

a. Show that if $\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$preserves length and angles, then the two column vectors $\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}$and $\stackrel{\mathbf{鈫�}}{\mathbf{w}}$of Amust be perpendicular unit vectors.

b. Write the first column vector of Aas $\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}\mathbf{=}\left[\begin{array}{l}a\\ b\end{array}\right]$ ; note that ${\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$, since $\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}$is a unit vector. Show that for a given $\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}$there are two possibilities for $\stackrel{\mathbf{鈫�}}{\mathbf{w}}$, the second column vector of A. Draw a sketch showing $\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}$and the two possible vectors $\stackrel{\mathbf{鈫�}}{\mathbf{w}}$. Write the components of $\stackrel{\mathbf{鈫�}}{\mathbf{w}}$in terms of a and b.

c. Show that if a linear transformation T fromR²to R² preserves length and angles, then Tis either a rotation or a reflection (about a line). See Exercise 17.

Step by step solution

01

Compute the vectors

Consider the matrix,

$A=\left[\raisebox{1ex}{$\stackrel{鈫�}{蠀}$}\!\left/ \!\raisebox{-1ex}{$\stackrel{鈫�}{w}$}\right.\right]$

Consider the vector

$\begin{array}{c}A\stackrel{鈫�}{蠀}=\stackrel{鈫�}{蠀}\\ 鈬�A\left[\begin{array}{l}1\\ 0\end{array}\right]=\stackrel{鈫�}{蠀}\\ 鈭�\stackrel{鈫�}{蠀}=\left[\begin{array}{l}1\\ 0\end{array}\right]\end{array}$$\begin{array}{c}A\stackrel{鈫�}{蠀}=\stackrel{鈫�}{蠀}\\ 鈬�A\left[\begin{array}{l}1\\ 0\end{array}\right]=\stackrel{鈫�}{蠀}\\ 鈭�\stackrel{鈫�}{蠀}=\left[\begin{array}{l}1\\ 0\end{array}\right]\end{array}$

Consider the vector,

$\begin{array}{c}A\stackrel{鈫�}{w}=\stackrel{鈫�}{w}\\ 鈬�A\left[\begin{array}{l}0\\ 1\end{array}\right]=\stackrel{鈫�}{w}\\ 鈭�\stackrel{鈫�}{w}=\left[\begin{array}{l}0\\ 1\end{array}\right]\end{array}$

As $\stackrel{鈫�}{蠀}=\left[\begin{array}{l}1\\ 0\end{array}\right],\stackrel{鈫�}{w}=\left[\begin{array}{l}0\\ 1\end{array}\right]$, thus, the matrix is perpendicular.

Therefore, the vectors of A must be perpendicular to each other.

02

Compute the possible vectors

Rotating by an angle $90掳$in either clockwise or anticlockwise direction a vector $\stackrel{鈫�}{蠀}$, we obtain the components of vector$\stackrel{鈫�}{w}$.

There are two possibilities.

$\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& 鈭�1\\ 1& 0\end{array}\right],\stackrel{鈫�}{蠀}=\left[\begin{array}{l}鈭�b\\ a\end{array}\right]鈬�\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& 鈭�b\\ a& 0\end{array}\right]$

$\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& 1\\ 鈭�1& 0\end{array}\right],\stackrel{鈫�}{蠀}=\left[\begin{array}{l}b\\ 鈭�a\end{array}\right]鈬�\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& b\\ 鈭�a& 0\end{array}\right]$

03

Check for the rotation and reflection property

Consider the matrix possibilities from the above step.

$A=\left[\begin{array}{cc}a& 鈭�b\\ b& a\end{array}\right]$

The above matrix represents that, the linear transformation has rotation characteristics.

04

Step 4:Final answer

a. The matrix Ahas perpendicular unit vectors,$\stackrel{鈫�}{蠀}=\left[\begin{array}{l}1\\ 0\end{array}\right],\stackrel{鈫�}{w}=\left[\begin{array}{l}0\\ 1\end{array}\right]$

b. $\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& 鈭�b\\ a& 0\end{array}\right]$and $\stackrel{鈫�}{w}=\left[\begin{array}{cc}0& b\\ 鈭�a& 0\end{array}\right]$are the possible components of the vector.

c. The linear transformation T has both rotation and reflection properties.

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