Let V be a finite dimensional vector space over C and let T be a linear operator on V. Suppose there exist non-zero vectors a and 3 in V such that T(a) cß, T(B) = c₂a. Prove that √cic2 or √c1c2 is an eigenvalue of T.

2. Let |1), |2), |3) be an orthonormal basis for a three dimensional vector space V. Find the eigenvalues and eigenvectors of the operator A = (|1) + |2) + |3))({1| + (2| + (3) Make sure to choose the eigenvectors so that they are orthonormal.

4 Let W = span(vV1, V2) where vi = and v2 2 . Then: (a) Find the orthogonal decomposition of the vector x = 3 (b) Find an orthogonal basis for W- (remember to justify your answer).

Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.

Let T : M2×2(R) → M2×2(R) be the linearoperator given byTa bc d =16a 5b + 11c11b + 5c 16d,where M2×2(R) denotes the vector space of 2 × 2-matrices over R.Let B =" 1 00 0 ,0 10 0 ,0 01 0 ,0 00 1 #denotethe standard basis of M2×2(R).(a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e.P[T]B(λ). Find the eigenvalues of [T]B and determine the algebraic multiplicity of each eigenvalue.(Hint: Eigenvalues are −6 and 16.)(b) Determine a basis for each eigenspace E[T]B(λ) of [T]B and determine the geometric multiplicity of each eigenvalue.(c) Use parts a and b to determine the characteristic polynomialPT (λ) of T, eigenvalues of T and a basis for each eigenspaceET (λ) of T. Show that eigenvectors of T form a basis B0forM2×2(R).(d) Find the representation [T]B0 of T relative to the basis B0. Use[T]B0 to verify that T is diagonalizable and T is an isomorphism.

Exercise 2.2.3 Let c₁=2i, c₂ = 1+2i, and A = 1-i 2+2i 3 4+i Verify Properties (vi) and (viii) in showing 2x2 is a complex vector space.

Let V be a finite-dimensional vector space over F with T in L(V,V) invertible and lambda in F \ {0}. Prove that lambda is an eigenvalue for T if and only if lambda-1 is an eigenvalue for T-1.

Find an orthonormal basis for the column space of A: A= 1 -2 1 0 1 1 1 3 and B= -4 -3 3 0