Let each of the following matrices represent an active...

Let each of the following matrices represent an active transformation of vectors in the (x, $y$ ) plane (axes fixed, vectors rotated or reflected). As in Example $3,$ show that each matrix is orthogonal, find its determinant, and find the rotation angle, or find the line of reflection. $$\frac{1}{\sqrt{2}}\left(\begin{array}{rr}1 & 1 \\-1 & 1\end{array}\right)$$

Let each of the following matrices represent an active transformation of vectors in the (x, $y$ ) plane (axes fixed, vectors rotated or reflected). As in Example $3,$ show that each matrix is orthogonal, find its determinant, and find the rotation angle, or find the line of reflection.
$$\frac{1}{\sqrt{2}}\left(\begin{array}{rr}1 & 1 \\-1 & 1\end{array}\right)$$

Key Concepts

Orthogonal Matrices

An orthogonal matrix is a square matrix whose columns and rows form an orthonormal set. This property ensures that the transformation preserves the length of vectors and the angles between them. In geometric terms, orthogonal matrices represent rotations and reflections, where the inverse of the matrix is simply its transpose.

Determinant in Transformations

The determinant of a transformation matrix provides information about the effect on area and orientation. For matrices representing rotations, the determinant is +1, indicating that both the area and orientation are preserved. In contrast, matrices with a determinant of -1 represent reflections, which reverse orientation while preserving area.

Rotation Matrices

Rotation matrices in the plane are defined in terms of sine and cosine of an angle. They transform vectors by rotating them around the origin without changing their lengths. The standard form of a rotation matrix in two dimensions is often given as [[cos?, sin?], [-sin?, cos?]], and comparing a given matrix to this form allows one to determine the rotation angle.

Active Transformations

Active transformations refer to operations that move or change the physical vectors or objects in space while keeping the coordinate system fixed. This contrasts with passive transformations, where the objects remain unchanged but the coordinates of the objects change due to a different choice of basis or reference frame.

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