課程名稱︰工程數學
課程性質︰生機系必修
課程教師︰顏炳郎
開課學院：生農學院
開課系所︰生機系
考試日期（年月日）︰108.11.26
考試時限（分鐘）：80
試題 :
1. Compute F x G where F = i + 4j - 3k, G = 2I - 6j + 5k. (10%)
2. Find the projections v = 5i + j - 2k on u = 3j - 8k. (10%)
3. Use the Gram-Schmidt process to find an orthogonal basis. The basis vector
┌ 3┐ ┌-5┐ ┌0┐
are {v1, v2, v3} = {│ 4│,│ 3│,│6│}. (10%)
└-2┘ └ 1┘ └7┘
4. Let S be the subspace of R^5 spanned by <2,1,-1,0,0>, <-1,2,0,1,0> and
<0,1,1,-2,0>. Find the vector in S closet to <4,3,-3,4,7>. (10%)
5. Evaluate the determinant. (Each 5%)
│ 3 4 -3│ │ 3 4 0 6│
(a) │-5 6 2│ (b) │ 0 1 4 7│
│ 7 -2 8│ │ 0 -2 1 9│
│-5 3 5 4│
6. Find the inverse of the matrix A. (Each 5%)
┌-2 0 -5┐
(a) A = ┌7 5┐ (b) A = │ 1 -4 1│
└3 -4┘ └ 2 3 2┘
7. Use a matrix inverse to find the unique solution of the system. (10%)
6x1 + x2 + 2x3 = 3
x1 - x2 + 4x3 = 1
-2x1 + 2x2 - 3x3 = 2
8. Use the reduced coefficient matrix to write a general solution. (10%)
-3x1 + x2 - x3 + x4 + x5 = 0
x2 + x3 + 4x5 = 0
- 3x3 + 2x4 + x5 = 0
9. Use the reduced coefficient matrix to write a general solution. (10%)
7x1 - 3x2 + 4x3 = -7
2x1 + x2 - x3 + 4x4 = 6
x2 - 3x4 = -5
10. Solve the system using Cramer's rule. (10%)
x1 + x2 - 3x3 = 0
x2 - 4x3 = 0
x1 - x2 - x3 = 5