[試題] 98下 馮世邁 線性代數 第一次小考
課程名稱︰線性代數
課程性質︰系必修
課程教師︰馮世邁
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰99年3月25日
考試時限（分鐘）：50 min
是否需發放獎勵金：是!感謝!
試題 :
1.Let the 3x5 matrix A and the vector b be respectively defined by
[ 1 0 -3 -1 -2] [1]
A = [a1 a2 a3 a4 a5]= [ 2 -1 -8 -1 -5] , b = [0]
[-1 1 5 1 4] [2]
(a)(20%) Find a lower triangular L and an upper triangular matrix U such
that [A b] = LU.
(b)(10%) Find the reduced row echelin form [R c] of [A b].
(c)( 5%) What are the rank and nullity of A?
(d)(10%) Find the general solution to Ax = b in vector form.
(e)( 5%) Find the general solution to Ax = 2b in vector form.
(f)( 5%) Choose three column vectors from A to form a 3x3 matrix A' so
that A'x = b is inconsistent.
(g)( 5%) Let S = {a1, a2, a3, a4, a5}. Find a linearly independent subset
S' of S such that Span S' = Span S.
2.(20%) Find the inverse of B = [a1 a2 a4] where a(i) are the column vectors
defined in Porblem 1.
3.(10%) Let S = {u1, u2, ..., u(k)} be a linearly independent subset of R^n.
Show that every vector in the span of S can be uniquely written as
a linear combination of the vectors in S.
4.(10%) Let A be a 3x4 matrix with reduced row echelon form R. And the rand
of A is 2. Find the reduced row echelon form of
(a) R(Transpose)
(b) the 6x4 matrix [A]
[A]
(You may express your answer in terms of A, R, O, and I.)
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