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題組內容

1. Consider the following system of linear equations.

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

其他申論題

2. R⊃S /∴ (P∨R)⊃(Q∨S)

(2) /∴ (x)(P⊃P)

(3) 1. (∃x)( y) Lxy /∴ ( y) (∃x) Lxy

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.

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