94年 - 94 淡江大學 轉學考 線性代數#56042

科目：轉學考-線性代數 | 年份：94年 | 選擇題數：0 | 申論題數：14

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所屬科目：轉學考-線性代數

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申論題 (14)

【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.
(b) (5 points) Sliow that L is invertible(可逆）.
(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.
【已刪除】8. (10 points) Let v1 and v2 be eigenvectors of a linear transformation T ： V →V with corresponding eigenvalues λ1 and λ2 respectively. Prove that, and v2 are indepeudent(線性獨立).
9. (10 points) Let P and Q be n x n matrices. We say P is similar to Q if there exists an invertible n x n matrix C such that C-lPC = Q. Prove that similar square matrices have the same eigenvalues with the same algebraic imilUplickies(代數重根數)•
【已刪除】10. (10 points) Let A be an n x n matrix such that Ax • Ay = x • y for all vectors x and y in . Show that A is an orthogonal matrix(正交矩陣).