Consider the linear transformation T: Rn → Rn whose matrix A relative to the standard basis is given.A = 4 1 −2 7(a) Find the eigenvalues of A. (Enter your answers from smallest to largest.)(𝜆1, 𝜆2) =  5,6 (b) Find a basis for each of the corresponding eigenspaces.B1  =  ⟨1,1⟩  B2  =  ⟨1,2⟩ (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b).

Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix.−1 32  − 121(a) the characteristic equationλ2−14​=0 (b) the eigenvalues (Enter your answers from smallest to largest.)(𝜆1, 𝜆2) =  −12​,12​ a basis for each of the corresponding eigenspacesx1  =  ⟨1,1⟩ x2  =  ⟨3,1⟩

Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6,    T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 =  v1​−v2​ 2 1 +  2v2​−v1​ 1 1(b) This implies the following.T(v) = T v1 v2= 2v1​ v1​+6v2​ = v

Find the linear transformation T : ℝ3 → ℝ2 that has the values given below on the given basis.T 1 00 = 3 8,    T 1 10 = 2 6,    T 1 11 = 7 5(a) First write a general vector v as a linear combination of the basis vectors.

(a) Find the eigenvalues, eigenvectors and eigenspaces forA =1 0 53 6 02 0 10(b) Calculate the determinant and trace of A directly and using eigen-values.

For each of the following matrices, and eigenvalue λ has been given.[ 0 4−1 5], λ = 4(a)[2 22 −1], λ = −2(b)3 1 −10 1 24 2 0, λ = 3(c)Find one eigenvector corresponding to the given eigenvalue