6. (15pts) Let V be a fnite dimensional vector space over C. Let T : V →Vbe a linear map and

let W be an T-invariant subspace of V. Suppose that U₁, U₂, , U_n are eigenvectors in V of T

corresponding to distinct eigenvalues λ₁,  λ₂, , λ_n in C. Prove that if u₁ +u₂+...+

then. (Hint: induction on n)

a. (5pts) Find the reduced row echelon form of its augmented matrix (A|b).

b. (10pts) Explicitly find the solution set K of this linear system, including a particular solution and a basis for its homogeneous solution set KH.

2. (15pts) Consider the following matrix Find a diagonal matrix D and an invertible matrix Q such that If it is impossible, give a convincing reason.

a. (5pts) Find the characteristic polynomial of T.

b. (5pts) Find the minimal polynomial m(x) of T. Explain why your answer is indeed  minimal.Hint: by defnition, m(x) is the monic non-zero polynomial with smallest degree such that m(T) is the zero map onR4.

4. (10pts) Let V = C3 be equipped with the standard inner product ＜, ＞. Let f: V → C be the unique linear map defined by In other words, f is an element in V*. Find the unique vector z V such that 

5. (15pts) Let W be the subspace of R4 spanned by the following vectors ข1=(1,-1,0,0) ข2=(0,1,-1,0) ข3=(0,0,1,-1) Apply the Gram-Schmidt algorithm to obtained an orthogonal basis for W.

7. (10pts) Let V = Rnbe equipped with the standard inner product ＜, ＞. Suppose that is a set of non-zero orthogonal vectors in V (m≤ n). Show that S is linearly independent.

8. (10pts) Let V and W be finite dimensional vector spaces and let T : V→ W be a linear map. Recall that there exists a unique linear map Tt : W*→ V*, called the transpose of T, where V* and W* are the dual spaces of V and W, respectively. Consider V = R2 and W = R3. Let T : V → W be the linear map defined by T(a,b)=(3a+6,a-26,2a-6)  Then there exists a unique linear map Tt : W* →V* as above. Define by Hint: you need to either explicitly write down Tt(θ)(a,b) for any (a,b), or simply write down a 2 x 1 matris representing Tt(θ).

Problem 6 :Let A and B be elements in (C). Suppose that AB - BA = c⋅(A - B)  for some non-zero c ∈ C. Prove that there exists an invertible matrix P ∈ (C) such that AP and BP are upper-triangular matrices with the same diagonal entries.

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