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：管理學院
開課系所︰工管系
考試日期（年月日）︰108/11/06
考試時限（分鐘）：100分鐘
試題 :
（以下＊代表向量，等於課本上的粗體、老師筆記中的底線。
（ε代表屬於）
1. Let A and its reduced row echelon form be respectively given by:
┌ 1 0 -3 -1 4┐ ┌ 1 0 -3 0 3 ┐
A = │ 2 -1 -8 -1 9│ R = │ 0 1 2 0 -2 │
│-1 1 5 0 -5│ │ 0 0 0 1 -1 │
└ 0 2 2 1 -3┘ └ 0 0 0 0 0 ┘
(a) Find a basis for: (i)Col A (ii)Null A (iii)Row A
(b) Verify whether the following set is a basis of Null A or not.
{ ┌ 3 ┐ ┌ -3 ┐ }
{ │-2 │ │ 2 │ }
S = { │ 2 │ │ 1 │ }
{ │ 1 │ │ 2 │ }
{ └ 1 ┘ , └ 2 ┘ }
2. Find the determinant of the following matrix:
┌ 1 -2 4 4 ┐
│ 0 0 4 -2 │
│ 0 1 3 -3 │
└ 4 -4 4 15 ┘
3. Let A = [a1* a2* a3* a4*] be a 4*4 matrix with reduced row echelon form R.
Suppose that det A = -3. Find the determinants of the following matrices:
(a) R (b) A' = [a4* a3* a2* a1*] (c) -A (d) [2A 3A]
[4A 4A]
4. Let T:R^4->R^3 be the linear transfromation defined by
( ┌ x1 ┐ ) ┌ x1 - x2 ┐
( │ x2 │ ) │ 2x2 + 2x3+ 2x4 │
T = ( │ x3 │ ) = │ 2x1 - 4x2 - 2x3 - x4 │
( └ x4 ┘ ) └ 2x1 - 4x2 - 2x3 - x4 ┘
(a) Find the standar matrix of T.
(b) Is T one-to one?
(c) Is T onto?
5. Let V be a subspace of R^n and dim V = k. Let u be an n*1 vector and u !ε V. Define
W = {wεR^n : w* = v* + cu* for some v*εV amd some scalar cεR }.
(a) Prove that W is a subspace of R^n.
(b) What is the dimension of W? (Explain your answer.)
6. Determine by inspection whether the following sets are linearly dependent or linearly
independent. (Explain your answer.) A correct answer without a correct explanation get 2%.
{ ┌ 1 ┐ ┌ 1 ┐ ┌ 1 ┐ }
{ │ 1 │ │ 0 │ │ 0 │ }
(a) S1 = {u1*, u2*, u3*} = { │ 2 │ │ 2 │ │ 2 │ }
{ │ 0 │ │ 1 │ │ 0 │ }
{ │ 1 │ │ 1 │ │ 1 │ }
{ └ 1 ┘, └ 0 ┘ , └ 1 ┘ }
{ ┌ 9 ┐ ┌ 4 ┐ ┌ 8 ┐ ┌ 5 ┐ ┌ 1 ┐ }
{ │ 7 │ │ 5 │ │ 3 │ │ 9 │ │ 2 │ }
(b) S2 = {v1*, v2*, v3*} = { │ 6 │ │ 6 │ │ 2 │ │ 6 │ │ 3 │ }
{ │ 0 │ │ 0 │ │ 0 │ │ 0 │ │ 5 │ }
{ └ 0 ┘, └ 0 ┘ , └ 0 ┘, └ 0 ┘, └ 4 ┘ }
{ ┌ 9 ┐ ┌ 5 ┐ ┌ 7 ┐ ┌ 1 ┐ }
{ │ 8 │ │ 7 │ │ 6 │ │ 2 │ }
(c) S3 = {w1*, w2*, w3*, w4*} = { │ 0 │ │ 0 │ │ 4 │ │ 3 │ }
{ └ 0 ┘, └ 0 ┘ , └ 0 ┘, └ 5 ┘ }
7. Let A be an m*n matrix and R be its reduced row echelon form. We know that there exists
an m*n matrix P such that PA=R. Suppose that rank A = m and let the pivot columns of A
be denoted by al1*, al2*, ... , alm* (i.e. al1* is the first pivot column, al2* is the
second pivot column, ... , alm* is the mth pivot column of A).
(a) Compute P・ali*, for i = 1, 2, ... , m.
(b) Prove that P is unique.