Question

Find a basis for the eigenspace corresponding to each listed eigenvalue of A below. A=[3 -1 0; 2 0 0; 4 2 5], λ=5,1,2

Answer 1

To find a basis for the eigenspaces of the given matrix A, one needs to solve (A - λI)v = 0 for each eigenvalue λ, find the corresponding eigenvectors, and ensure they are linearly independent to form a basis of the eigenspace.

Step-by-step explanation:

To find a basis for the eigenspace corresponding to each listed eigenvalue of the matrix A = [3 -1 0; 2 0 0; 4 2 5], where the eigenvalues are λ = 5, 1, 2, we need to solve the respective eigenvector equations given by (A - λI)v = 0, where I is the identity matrix, and v is an eigenvector corresponding to the eigenvalue λ.

For each eigenvalue, we subtract λI from A and then solve the resulting system of linear equations:

1. For λ = 5, we have (A - 5I)v = 0.
2. For λ = 1, we have (A - I)v = 0.
3. For λ = 2, we have (A - 2I)v = 0.

For each system, we reduce it to its row-echelon form and solve for the free variables to find the corresponding eigenvectors. The set of all independent eigenvectors for each eigenvalue forms the basis for its eigenspace.