Final answer:
The standard matrix A for the rotation of points in R² by –π5π/4 radians is calculated using cos and sin of the angle, resulting in A = [[√2/2, √2/2],[√2/2, -√2/2]].
Step-by-step explanation:
The student is seeking the standard matrix for a linear transformation that represents a rotation of points about the origin by –π5π/4 radians in R². To construct the standard matrix of transformation T, which rotates points, we use the trigonometric relations for rotating coordinates:
- x' = x cos θ + y sin θ
- y' = –x sin θ + y cos θ
As θ = –π5π/4 radians, cos(–π5π/4) = cos(225°) = –√2/2 and sin(–π5π/4) = sin(225°) = –√2/2. Plugging these values into the rotation formulas gives us the standard matrix for T:
A =
\[ \begin{bmatrix} \frac{√2}{2} & \frac{√2}{2} \\ \frac{√2}{2} & \frac{-√2}{2} \end{bmatrix} \]
The standard matrix A for the linear transformation T is a 2x2 matrix where the entries are determined by the rotation angle θ. Given θ = –π5π/4, the matrix is A = [[√2/2, √2/2],[√2/2, -√2/2]].