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107年 - 107 東吳大學_暑假轉學生招生考試_哲學系二年級：邏輯#105305
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四、請用有效的推論規則證明下列論證（從前提推導出結論）。
(1)
2. R⊃S /∴ (P∨R)⊃(Q∨S)
相關申論題
(2) /∴ (x)(P⊃P)
#446854
(3) 1. (∃x)( y) Lxy /∴ ( y) (∃x) Lxy
#446855
a. (5pts) Find the reduced row echelon form of its augmented matrix (A|b).
#446856
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.
#446857
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.
#446858
a. (5pts) Find the characteristic polynomial of T.
#446859
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.
#446860
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
#446861
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.
#446862
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)
#446863
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