Question

Find the matrix in the linear transformation, where x and y are Cartesian coordinates. Find the eigenvalues and eigenvectors and explain their geometrical meaning in the following transformations:

(i) Counter-clockwise rotation through the angle θ about the origin.
(ii) Reflection about the axis.
(iii) Orthogonal projection of v onto the axis.
(iv) Orthogonal projection of v onto the plane P.

Answer 1

Eigenvalues and eigenvectors have geometric interpretations in linear transformations such as rotation, reflection, and orthogonal projection, indicating scaling factors and invariant directions respectively. (ii) Reflection about the axis is correct answer.

Step-by-step explanation:

The subject of this question involves understanding the geometrical meaning of eigenvalues and eigenvectors in various linear transformations in two-dimensional Cartesian coordinates.

The transformations include counter-clockwise rotation, reflection about an axis, and orthogonal projections onto an axis and onto a plane. Eigenvalues and eigenvectors are fundamental concepts in linear algebra related to these transformations and have important geometric interpretations.

For a counter-clockwise rotation through angle θ about the origin, the transformation matrix is a 2x2 matrix consisting of cos(θ) and sin(θ) in a certain arrangement.

The eigenvalues in this case are complex and convey the rotation, while the eigenvectors give the directions that are invariant under the transformation. Reflection and orthogonal projection can also be described by transformation matrices, with eigenvalues indicating scaling (if any) and eigenvectors showing invariant directions or planes.

In summary, eigenvalues indicate the factor by which the transformation scales vectors, and eigenvectors provide the directions that remain unchanged under the transformation.