Call \(\mathrm{R}(\theta)\) the rotation matrix: $$ \mathrm{R}(\theta)=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$ Show that \(\mathrm{R}(-\theta)\) rotates by \((-\theta)\). What can you say about \(\mathrm{R}(-\theta) w\) when \(w=\mathrm{R}(\theta) z\) ?

Trigonometric identities are fundamental relationships between trigonometric functions. These identities allow for the simplification and transformation of trigonometric expressions. In this exercise, we particularly use the identities for negative angles: \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta).\)
These identities help us transition from \( \mathrm{R}(\theta) \) to \( \mathrm{R}(-\theta) \). By recognizing that the cosine function is even and the sine function is odd, we can easily calculate the rotation matrix for negative angles.
Understanding trigonometric identities is crucial not only for matrix transformations but also for a wide range of mathematical and engineering applications.

Matrix multiplication is a key operation in linear algebra. It consists of multiplying two matrices by taking the dot product of rows and columns. For the rotation matrices \( \mathrm{R} (-\theta) \) and \( \mathrm{R} (\theta) \), matrix multiplication helps us determine the resultant transformation when applying two successive rotations:
\[ \mathrm{R}(-\theta) \mathrm{R}(\theta) = \left[\begin{array}{cc} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{array}\right] \left[\begin{array}{cc} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{array}\right] \]
Through matrix multiplication, each entry of the resulting matrix is calculated by summing the products of corresponding elements from the rows of the first matrix and columns of the second matrix. This leads to the identity matrix, indicating that\( \mathrm{R}(-\theta) \mathrm{R}(\theta) \) effectively cancels out any rotation applied by \( \mathrm{R}(\theta) \).
Practice calculating these products to become comfortable with matrix multiplication.

The identity matrix is a special matrix that acts as the multiplicative identity in matrix multiplication. This means any matrix multiplied by the identity matrix remains unchanged. For a 2x2 matrix, the identity matrix \( I \) is:
\[ I = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] \]
In this exercise, we see that \( \mathrm{R}(-\theta) \mathrm{R}(\theta) \) results in the identity matrix. Thus, applying \( \mathrm{R}(-\theta) \) after \( \mathrm{R}(\theta) \) returns any vector to its original position:
\[ \mathrm{R}(-\theta) \mathrm{R}(\theta) = I \]
Recognizing the identity matrix helps understand how complex transformations can be neutralized. It's essential in linear algebra, as it assures us that combined inverse operations will result in no change.

Angle rotation is a fundamental concept in geometry and linear transformations. Rotating a vector in the plane by an angle \( \theta \) involves altering its direction while preserving its magnitude. The standard rotation matrix \( \mathrm{R}(\theta) \) is:
\[ \mathrm{R}(\theta) = \left[\begin{array}{cc} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{array}\right] \]
This matrix effectively rotates any vector by angle \( \theta \). Conversely, \( \mathrm{R}(-\theta) \) rotates a vector by \( -\theta \), which is essential in understanding inverse transformations. Using rotation matrices, we can describe how any point or vector moves and settles in a coordinate plane upon rotation.
Understanding this concept helps in various fields such as computer graphics, robotics, and any domain that involves transformations of objects in space.