[試題] 108-1 馮世邁 線性代數 小考一
課程名稱︰線性代數
課程性質︰工管系科管組必修
課程教師︰馮世邁
開課學院：管院
開課系所︰工管系
考試日期（年月日）︰108.10.16
考試時限（分鐘）：
試題 :
1. Let A and b be respectively defined by
┌ 1 -1 0 2 0┐ ┌ 2 ┐
A = [a1 a2 a3 a4 a5] = │-1 1 0 2 4│ , b = │ 6 │
│ 2 -3 -1 2 -3│ │ 3 │
└-3 2 -1 0 5┘ └ 9 ┘
(a) (20%) Find the reduced row echelon form of [A b]
(b) (6%) What are the rank and nullity of A?
(c) (10%) Find the general solution to Ax = b in vector form.
(d) (6%) Write each non pivot column of A as a linear combination of the piv-
ot columns of A.
(e) (5%) (BONUS) Let A' = [a2 a1 a3 a4 a5] and b' = b - 2.847a4. Without usi-
ng the Gaussian elimination, find the general solution to A'x = b'.
2. (20%) Find the inverse of the following matrix:
┌ 0 2 -1 ┐
│ 1 -1 2 │
└ 2 -1 3 ┘
3. (18%) Consider the following system of linear equations:
x1 + x2 + x3 = 1
x1 + 3x3 = -2 + s
x1 - x2 + rx3 = 3
(a) For what values of r and s is this system of linear equations inconsistent
(b) For what values of r and s does this system of linear equations have infi-
nitely solutions?
(c) For what values of r and s does this system linear equations have a unique
solution?
4. (20%) Let A = [a1 a2 ... an] be an mxn matrix and R be its reduced row ech-
elon form, Let b in R^m. Label the following statements as being true
or false. (No explanation is needed. Each correct answer get 4% and
each wrong answer gets 0%).
(a) Ax = b is consistent if and only if rank A = rank[A b].
(b) If m > n, then Ax = b is inconsistent.
(c) Ax = b is consistent if and only if
Span {a1, a2, ..., an, b} = Span {a1, a2, ..., an}.
(d) If m < n, then {a1, a2, ..., an} is linearly dependent.
(e) Let P be an mxm matrix. If PA = R, then P is invertible.
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