Final answer:
To find the matrix of the linear transformation T in the standard basis, apply T to each of the basis vectors to determine the matrix columns. Then, to find the eigenvalues and eigenspaces of the matrix, solve the characteristic equation and use the eigenvalues to solve for the eigenvectors.
Step-by-step explanation:
The question relates to linear algebra and involves finding a matrix corresponding to a linear transformation and then determining its eigenvalues and eigenspaces. This falls within the scope of advanced mathematics, typically covered at the college level. To find the matrix of the linear mapping T with respect to the standard basis, we apply the transformation to each of the basis vectors and use the results as the columns of the matrix. Given T(x, y, z) = (2x – y + 2, x + 2, x – y + 2z), applying T to (1,0,0), (0,1,0), and (0,0,1) yields:
- T(1, 0, 0) = (2, 1, 1), which forms the first column.
- T(0, 1, 0) = (–1, 0, –1), which forms the second column.
- T(0, 0, 1) = (2, 2, 2), which forms the third column.
The resulting matrix A is:
A = [ [2, –1, 2], [1, 0, 2], [1, –1, 2] ]
To find the eigenvalues of A, we solve the characteristic equation det(A – λI) = 0. This involves finding the determinant of the matrix resulting after subtracting λ times the identity matrix from A and setting this determinant equal to zero. The solutions for λ are the eigenvalues of A. Once the eigenvalues are known, we can find the eigenspaces by solving the equation (A – λI)x = 0 for each eigenvalue λ, where I is the identity matrix and x is an eigenvector.