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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.

其他申論題

(2) /∴ (x)(P⊃P)

(3) 1. (∃x)( y) Lxy /∴ ( y) (∃x) Lxy

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.

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.

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, , Un are eigenvectors in V of T corresponding to distinct eigenvalues λ1, λ2, , λn in C. Prove that if u1 +u2+...+ then. (Hint: induction on n)

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.

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