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課程名稱︰線性代數
課程性質︰
課程教師︰馮蟻剛
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2008/03/20
考試時限（分鐘）：50
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
1. Determine whether the given vector v is in the span of
[ -1 ] [ 1 ] [ 1 ]
S = { [ 5 ] ,[ 3 ] ,[ -1 ] }
[ 2 ] ,[ 4 ] ,[ 1 ]
T
(a.) v = [ 5 3 11 ] (20 %)
T
(b.) v = [ 1 1 2 ] (10 %)
If so, write v as linear combination of vectors in S.
[ 1 -3 0 5 3 ]
2. Suppose that the reduced row echelon form of A is R = [ 0 0 1 2 -2 ]
[ 0 0 0 0 0 ]
, determine A if the first and third colums of A are
[ 1 ] [ 2 ]
a1 = [ -1 ] and a3 = [ 0 ], respectively. (30 %)
[ 2 ] [ -1 ]
3. Let A and B be n * n matrices. We say that A is similar to B if
B = P^(-1) A P for some invertible matrix P.
(a.) Prove that if A is similar to B and B is similat to C,
then A is similar to C.
(b.) Suppose that A is similar to B. Prove that if A is invertible,
then B is invertible, and A^(-1) is similar to B^(-1). (10%)
4. (a.) Prove that if A is a m * n matrix and B is an n * p matrix,
then rank(AB) ≦ rank (B).
( Hint : Prove that if the k-th column of B is not a pivot column of
B, then the k-th column of AB ois not a pivot column of AB.) (10 %)
(b.) Prove that is A is a m * n matrix and P is an invertible m * m
matrix, then rank(PA) = rank(A).
( Hint: Apply 4(a.) to PA and to P^(-1)(P A) ). (10%)
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