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

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.

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.

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).

題目一:【10分】 機械製造主要分為切削性加工與非切削性加工,切削性加工功能在於移除材料,可進一步 分為傳統切削加工法和非傳統切削加工法:至於非切削性加工則可進一步分為改變材料形 狀的成型加工法和連結材料的接合加工法。請將車削、鍛造、放電加工、鉚接、銑削、雷 射加工、铸造、電解加工、射出成型和電鍍10種加工法,依照上述傳統切削加工法、非 傳統切削加工法、成型加工法和接合加工法四大類分別歸類。

(一)說明熱作和冷作的主要差異。【5分】

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