b,c ### Linear Transformations and Diagonalization#### Problem 2: Diagonalization of TransformationsFor the following transformations, find a basis \( B \) such that the matrix representing \( T \) is diagonal.#### (a) Transformation in \(\mathbb{R}^2\)\[ T(x, y) = (3x + 2y, 2x + 3y) \]Find a basis \( B \) for which the matrix representing \( T \) is diagonal.#### (b) Transformation for Polynomial Coefficients\[ T(a_0 + a_1x) = a_0 - a_1x \]Find a basis \( B \) for which the matrix representing \( T \) is diagonal.#### (c) Transformation in \(\mathbb{R}^3\)\[ T(x, y, z) = (-2x + 2y - 3z, 2x + y - 6z, -x - 2y) \]For this transformation, show the steps of calculating the determinant until you obtain a simplified cubic polynomial. After getting the cubic, it can be written as:\[ -(\lambda + 3)^2(\lambda - 5) \]Continue solving from this form.In this problem, we are focusing on finding an eigenbasis for each transformation that will allow the matrix \( T \) to be expressed in a diagonal form. This often involves calculating the eigenvalues and eigenvectors for the given matrix representations of the transformations.

Name: b,c ### Linear Transformations and Diagonalization#### Problem 2: Diag
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: b,c ### Linear Transformations and Diagonalization#### Problem 2: Diagonalization of TransformationsFor the following transformations, find a basis \( B \) such that the matrix representing \( T \) is diagonal.#### (a) Transformation in \(\mathbb{R}^