Rotational Transformations A mapping T: ℝ^2 →ℝ^2 is given by T(𝐯)=𝐀 𝐯, where 1. 𝐀=[ cosθ

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Rotational Transformations A mapping T: ℝ^2 →ℝ^2 is given by...

Rotational Transformations A mapping $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is given by $T(\overrightarrow{\mathbf{v}})=\mathbf{A} \overrightarrow{\mathbf{v}}$, where $$ \text { 1. } \mathbf{A}=\left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] . $$ Show that $T$ rotates every vector $\overrightarrow{\mathrm{v}} \in \mathbb{R}^2$ counterclockwise about the origin through angle $\theta$. HINT: Express $\vec{v}$ using polar coordinates, $$ \overrightarrow{\mathbf{v}}=\left[\begin{array}{c} r \cos \alpha \\ r \sin \alpha \end{array}\right] . $$ and use the identities for $\cos (\theta+\alpha)$ and $\sin (\theta+\alpha)$.

Rotational Transformations A mapping $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is given by $T(\overrightarrow{\mathbf{v}})=\mathbf{A} \overrightarrow{\mathbf{v}}$, where
$$
\text { 1. } \mathbf{A}=\left[\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right] .
$$
Show that $T$ rotates every vector $\overrightarrow{\mathrm{v}} \in \mathbb{R}^2$ counterclockwise about the origin through angle $\theta$. HINT: Express $\vec{v}$ using
polar coordinates,
$$
\overrightarrow{\mathbf{v}}=\left[\begin{array}{c}
r \cos \alpha \\
r \sin \alpha
\end{array}\right] .
$$
and use the identities for $\cos (\theta+\alpha)$ and $\sin (\theta+\alpha)$.

Key Concepts

Rotation Matrix

A rotation matrix is a specific type of square matrix used to perform a rotation in Euclidean space. In two dimensions, it rotates vectors by a fixed angle while preserving the magnitude of the vector. The standard 2x2 rotation matrix includes trigonometric functions that handle the rotation through cosine and sine, ensuring that the transformation is length-preserving and angle-preserving.

Linear Transformation

A linear transformation refers to a function between vector spaces that preserves vector addition and scalar multiplication. When such transformations are represented by matrices, they provide a concrete method for altering vectors' orientation, magnitude, or both in a consistent and predictable manner, making them fundamental in linear algebra.

Polar Coordinates

Polar coordinates are a way of expressing a point in the plane using a distance from the origin and an angle relative to the positive x-axis. This representation is especially useful for problems involving rotations because it naturally separates the radial component from the angular component, simplifying the analysis of rotational effects on the vector.

Trigonometric Identities

Trigonometric identities, such as the angle sum formulas for sine and cosine, provide relationships that allow the expression of the sine or cosine of a sum of angles in terms of the sine and cosine of the individual angles. These identities are crucial in proving that the rotation matrix correctly rotates a vector by the specified angle, as they connect the polar angle representation of the vector with the new coordinates after rotation.

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