Question

In each case show that that \(T\) is either projection on a line, reflection in a line, or rotation through an angle, and find the line or angle. a. \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\frac{1}{5}\left[\begin{array}{c}x+2 y \\ 2 x+4 y\end{array}\right]\) b. \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\frac{1}{2}\left[\begin{array}{l}x-y \\\ y-x\end{array}\right]\) c. \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}-x-y \\\ x-y\end{array}\right]\) d. \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\frac{1}{5}\left[\begin{array}{c}-3 x+4 y \\ 4 x+3 y\end{array}\right]\) e. \(T\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}-y \\\ -x\end{array}\right]\) f. \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\frac{1}{2}\left[\begin{array}{l}x-\sqrt{3} y \\ \sqrt{3} x+y\end{array}\right]\)

Short answer

a. Projection onto line \(y=2x\). b. Reflection in line \(y=x\). c. Rotation by \(45^\circ\). d. Rotation by \(53.13^\circ\). e. Rotation by \(90^\circ\). f. Rotation by \(30^\circ\).

Step-by-step solution

Analyze Matrix Transformation a

Given the transformation: \( T\left[ \begin{array}{c} x \ y \end{array}\right] = \frac{1}{5} \left[ \begin{array}{c} x + 2y \ 2x + 4y \end{array} \right] \).Represent this transformation as a matrix multiplication: \[ T = \frac{1}{5} \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \].The matrix simplifies to: \[ \begin{bmatrix} 0.2 & 0.4 \ 0.4 & 0.8 \end{bmatrix} \].Check if \(T^2 = T\):- \( \begin{bmatrix} 0.2 & 0.4 \ 0.4 & 0.8 \end{bmatrix} \cdot \begin{bmatrix} 0.2 & 0.4 \ 0.4 & 0.8 \end{bmatrix} = \begin{bmatrix} 0.2 & 0.4 \ 0.4 & 0.8 \end{bmatrix} \).Since \( T^2 = T \), \(T\) is a projection. The projection is onto the line described by the vector \( \begin{bmatrix} 1 \ 2 \end{bmatrix} \).

Analyze Matrix Transformation b

Given the transformation: \( T\left[ \begin{array}{c} x \ y \end{array}\right] = \frac{1}{2} \left[ \begin{array}{c} x-y \ y-x \end{array} \right] \).Represent this transformation as a matrix multiplication: \[ T = \frac{1}{2} \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \].The matrix simplifies to: \[ \begin{bmatrix} 0.5 & -0.5 \ -0.5 & 0.5 \end{bmatrix} \].Check if \(T^2 = I\):- \( \begin{bmatrix} 0.5 & -0.5 \ -0.5 & 0.5 \end{bmatrix} \cdot \begin{bmatrix} 0.5 & -0.5 \ -0.5 & 0.5 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).Since \( T^2 = I \), \(T\) is a reflection. The line of reflection is \( y = x \).

Analyze Matrix Transformation c

Given the transformation: \( T\left[ \begin{array}{c} x \ y \end{array}\right] = \frac{1}{\sqrt{2}} \left[ \begin{array}{c} -x - y \ x - y \end{array} \right] \).Represent this transformation as a matrix multiplication: \[ T = \frac{1}{\sqrt{2}} \begin{bmatrix} -1 & -1 \ 1 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \].The matrix simplifies to: \[ \begin{bmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \].Check if it represents a rotation by evaluating eigenvalues:- Eigenvalues are \(\pm i\), which indicates a rotation of \(45^\circ\).

Analyze Matrix Transformation d

Given the transformation: \( T\left[\begin{array}{c}x \y\end{array}\right]=\frac{1}{5}\left[\begin{array}{c}-3x+4y\4x+3y\end{array}\right]\).Represent this transformation as a matrix multiplication: \[ T = \frac{1}{5} \begin{bmatrix} -3 & 4 \ 4 & 3 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \].The matrix simplifies to:\[ \begin{bmatrix} -0.6 & 0.8 \ 0.8 & 0.6 \end{bmatrix} \].This can be identified as a rotation since the determinant is \(1\), confirming it is an orthogonal transformation. Solving for the angle gives \( \theta = \arccos(0.6) \), approximately \(53.13^\circ\).

Analyze Matrix Transformation e

Given the transformation: \( T\left[ \begin{array}{c} x \ y \end{array}\right] = \left[ \begin{array}{c} -y \ -x \end{array} \right] \).Represent this transformation as a matrix multiplication: \[ T = \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \].Check the determinant and eigenvalues:- Eigenvalues \(\pm i\) confirm it's a rotation by \(90^\circ\).

Analyze Matrix Transformation f

Given the transformation: \( T\left[ \begin{array}{c} x \ y \end{array}\right] = \frac{1}{2} \left[ \begin{array}{c} x - \sqrt{3}y \ \sqrt{3}x + y \end{array} \right] \).Represent this transformation as a matrix multiplication: \[ T = \frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \ \sqrt{3} & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \]The matrix simplifies to: \[ \begin{bmatrix} 0.5 & -0.866 \ 0.866 & 0.5 \end{bmatrix} \].This matrix is a rotation matrix with angle \( \theta = 30^\circ \) as derived from \(\cos \theta\) and \(\sin \theta\).

Key concepts

Projection

Matrix transformations often include projections, which essentially map a vector onto another vector or line. Projections can be thought of as shadows cast by a vector onto a line or another vector. In the context of transformations, this is represented by a matrix that scales and aligns your original vector to the target line.

To verify if a transformation is a projection, ensure that the transformation matrix satisfies the equation \(T^2 = T\). This demonstrates that applying the transformation twice does not change the result, a key property of projections.

For example, consider the matrix:\[ \begin{bmatrix} 0.2 & 0.4 \ 0.4 & 0.8 \end{bmatrix} \]
This matrix represents a projection because repeatedly applying the matrix gives the same result as applying it once. This particular projection aligns vectors with the line determined by the vector \(\begin{bmatrix} 1 \ 2 \end{bmatrix}\).

• Projections decrease dimensionality, possibly reducing a 3D vector to lie entirely within a 2D plane.
• They can visualize components of a vector parallel or perpendicular to a line.
• Understanding projections is crucial in fields like computer graphics and physics, where direction and axis alignment matter.

Reflection

Reflection in matrices involves flipping a vector or shape across a specified line, much like viewing an image in a mirror. This concept is depicted mathematically by transformation matrices that ensure the operation \(T^2 = I\), with \(I\) being the identity matrix. This signifies that if you reflect twice, you return to the original vector or shape.

In this matrix transformation problem, for instance, consider the matrix:\[ \begin{bmatrix} 0.5 & -0.5 \ -0.5 & 0.5 \end{bmatrix} \]
This matrix indicates a reflection over the line \(y = x\). When applied, it changes the coordinates by reflecting each point through the line \(y = x\), making it a symmetric process.

• Reflections preserve lengths and angles leading to congruent figures.
• In 2D, reflections are defined with respect to a line, while in 3D, they're with respect to a plane.
• Common in optics (physics) where reflection properties are used to understand how angles of incidence equal angles of reflection.

Rotation

Rotations in transformation matrices involve turning a vector around a fixed point, typically the origin, by a specific angle. The terms \(\cos \theta\) and \(\sin \theta\) determine how much each coordinate is rotated, maintaining the original's magnitude.

A key trait of rotation matrices is their determinant equaling 1, confirming the rotation keeps vectors the same length but changes orientation. For example, consider the matrix transformation:\[ \begin{bmatrix} -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \]
This results in a 45-degree rotation, as indicated by its complex eigenvalues \(\pm i\).

• In 2D, a positive angle results in a counter-clockwise rotation while negative angles rotate clockwise.
• Rotations are fundamental in robotics where positioning mechanisms require precise rotations.
• Rotate functions and matrices are crucial in graphics to animate with smooth transitions between frames.