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題組內容

4. Let L :

be a linear transibnnation defined by L

(b) (5 points) Sliow that L is invertible(可逆）.

其他申論題

【已刪除】1. (5 points) Let A and B be n x n matrices. Is If so, prove it; if not, give a counterexample(反例）and state ujicler what conditions the equation is true.

【已刪除】2. (5 points) Let a, b and c be real numbeis(實數)such that abc ≠ 0. Prove that the plane ax + by -f- cz = 0 is a subspace(子空間)of

【已刪除】3. (10 points) Let T : be a linear transformatioa. If T([l,0,0]) = [-3,1], T([0，l,0]) = [4,-1], and T([0,-i, 1]) = [3,5], find T[-1,4,2]).

(a) (5 points) Find the standard matrix representation(標準矩陣表示式)of L.

(c) (5 points) Find a formula for L-1

【已刪除】5. (10 points) Let K be a vector space with basis(基底）

(a) (5 points) lYansfer this system as a matrix form Ax = b and write down A and b.

(b) (5 points) Solve the linear system by Cramer’s rule.

(a) (10 points) Find the characterisLic polynomial(特徵多項式),the real eigenvalues(特徵值）＞ and the coriesponding eigenvectors(特徵向簠）of A.

(b) (5 points) Find an invertible matrix C and a diagonal matrix(對角矩陣）D such that D 二 C-MC.

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