Final answer:
The question involves calculating eigenvalues and eigenvectors for given matrices represented by their characteristic polynomial. However, the matrices themselves are not provided. The process includes finding eigenvalues, solving for eigenvectors, and establishing a basis for each eigenvalue's eigenspace.
Step-by-step explanation:
The question pertains to linear algebra, specifically to eigenvalues and eigenvectors of matrices. For the given matrices A and B with the characteristic polynomial p(t) = t³ - 3t + 2, we need to find the eigenvalues and corresponding eigenspaces, denoted W_lambda, which are the bases for each eigenvalue lambda. To find a basis for W_lambda, we must solve the equation (A - lambda*I)x = 0 for matrix A and the same for matrix B, where lambda represents an eigenvalue and I is the identity matrix of the same size as A and B.
Unfortunately, the matrices A and B are not provided in the question. However, the process involves calculating the characteristic polynomial, finding the eigenvalues by solving for lambda in the equation p(lambda) = 0, and then, for each eigenvalue, finding a set of non-zero vectors that satisfy the eigenvector equation, which forms a basis for the corresponding eigenspace W_lambda.
To find the eigenvectors, substitute each lambda back into the equation (A - lambda*I)x = 0 and solve for the vector x. The solution set of this system will be the eigenspace, and you can extract a basis from this set.