3. Let T` : V → W be a linear transformation. V and W be inner product spaces. Show that TT and TT" has exactly the same non-zero eigenvalues. 4. Let L€ L(V.W), where V and Ware inner product spaces. Given orthonormal basis 8 = {u} of V_and_p = {ws of W. suppose we have the following information 2 7 x2 ? 12 6 dim(Range(L)) = 1; 2012 3 12 ? where? are unknown numbers. Determine the Schmidt decomposition of L.

Let a be a fixed unit length vector (A is in R^3). Define a linear transformation T: R^3 -> R^3 by T(x)=(a.x)a (a.x refers the dot product between a and x). If we can compute the eigenvalues of T to be 0 and 1, what would be the dimension of the eigenspaces corresponding with the found eigenvalues ? Is the standard matrix T diagonalizable ?

The unit vectors x in R2 and their images Ax under th e action of a 2 X 2 matrix A are drawn head-to-tail, as in Figure . Estimate th e eigenvectors and eigenvalues of A from each ''eigenpicture."

If A al for some real constant a, then any vector v is an eigenvector, and the origin 0 is a star point or proper node of the Ax. system æ' True False

Suppose the vectors u and v are a basis B for R^2 and T is a transformation from R^2 → R^2 that maps u to 3u, and v to u+3v. Let matrix A such that T(¯x) = Ax¯.  (a) Arguing geometrically, identify one eigenvalue λ for the matrix A, and describe the eigenspace in terms of u and v. What will be the geometric multiplicity of λ?  (b) Find such a matrix which is the B matrix for T.  (c) Find a relationship between the eigenvalues of A and the eigenvalues of B. Using the previous part’s answer, explain why there is only 1 eigenvalue for matrix A.  (d) What can be said about the algebraic multiplicity of the eigenvalue λ from the first part? Can matrix A be diagonalized?

Consider the linear transformation T: R^ →R" whose matrix A relative to the standard basis is given. [³ A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 12) = 4,5 B₂ 3-2 (b) Find a basis for each of the corresponding eigenspaces. B₁ = 1,2 128 1,1 = A' = 1 6 (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). X 5 5 1 2 ↓ 1

Construct a linear operator which has exactly two eigenvalue.

- find the eigenvectors of the diagonal matrix (A) J where. 2 5 0-12 2 4 10 A = and P=[^}; 2 1 5 4 ♡ 1

Find a linear transformation I having 1₁ = 1 as an an eigenvalue associated to the vector (1₁ 1₁0) and 1₂=5 as eigenvalue associated to the vectors (-1,1,0) and (0,0,1). What is the characteristic polynomial of T?

What are two linearly independent eigenvectors of A. Are those eigenvectors linearly independent? Is it a basis for R^2