推 djshen :真快..才剛考完 03/24 15:42
課程名稱︰工程數學-線性代數
課程性質︰必修
課程教師︰馮蟻剛
開課學院：電機學院
開課系所︰電機系
考試日期（年月日）︰2011.03.24
考試時限（分鐘）：55
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
T(A)表示為A的轉置矩陣 A(ij)表示A中的第i row 第j column。
1.Judge if the following statements are true or false.
Give a concise proof to each true statement, and a counterexample
to each false statement.
(a) If X and Y are invertible n*n matrices, then X+Y is also invertible.(20%)
(b) Every linear combination of the rows of a matrix B is also a linear
combination of the rows of the reduced row echelon form of B. (20%)
2. Suppose R is the reduced row echelon form of an m*n matrix A with rank
r, and the reduced echelon form of T(R) is the n*m matrix Q=q(ij).
Show that q(ii)=1 for i = 1,...,r, and q(ij)=0 elsewhere. (25%)
Use the above result to deduce first that rank T(A) 小於等於 rank A.
Then deduce further that rank T(A)= rank A. ( 5%)
(Hint: Note there exist an invertible matrix P and elementary matrices
E1,...,Ek such that A = PR and Ek...E1T(R)T(P)=QT(P) )
3. For
1 0 -1 -1
0 -1 1 -9
S={[-1],[ 2],[ 1],[ c]},find value(s) of c making S linearly dependent.(30%)
1 1 0 2
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沒有伴隨著痛苦的教訓是沒意義的。
人如果不犧牲一些東西，就無法得到任何東西。
但是當超越了障礙，並且把得到的東西變成屬於自己的東西時...
人應該就能夠得到無法取代的鋼之心靈吧。
<Fullmetal Alchemist>
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