Question

Find a basis of the eigenspace associated with the eigenvalue -2 of the matrix A=⎣⎡​−20−11​0−200​−20−11​−201−1​⎦⎤​ A basis for this eigenspace is {[],[} If v1​=[−3−2​] and v2​=[−20​] are eigenvectors of a matrix A corresponding to the eigenvalues λ1​=1 and corresponding to the eigenvalues λ1​=1 and λ2​=−4, respectively, then A(v1​+v2​)=[] and A(2v1​)=[]

Answer 1

To find a basis of the eigenspace associated with the eigenvalue -2 of the matrix A, we need to find the null space of the matrix (A - (-2)I). A basis for the eigenspace associated with the eigenvalue -2 is { [1, 0, -1] }. A(v1 + v2) = [3, -2, -4] and A(2v1) = [-6, -4, -2].

Step-by-step explanation:

To find a basis of the eigenspace associated with the eigenvalue -2 of the matrix A, we need to find the null space of the matrix (A - (-2)I), where I is the identity matrix. The null space of a matrix is the set of vectors that, when multiplied by the matrix, result in the zero vector.

After performing the necessary calculations, we find that the null space of the matrix (A - (-2)I) is spanned by the vector [1, 0, -1]. Therefore, a basis for the eigenspace associated with the eigenvalue -2 is { [1, 0, -1] }.

To evaluate A(v1 + v2), we substitute the given values of v1 and v2 into the matrix A and perform the necessary computations. Similarly, to evaluate A(2v1), we multiply the vector v1 by 2 and substitute the result into the matrix A. After performing the calculations, we obtain the following results:

A(v1 + v2) = [3, -2, -4]

A(2v1) = [-6, -4, -2]