课程名称︰工程数学一
课程性质︰必修
课程教师︰刘进贤
开课学院：工学院
开课系所︰土木工程学系
考试日期（年月日）︰103/11/10
考试时限（分钟）：130分钟
试题 :
The First Mid-Term Examination of Engineering
Mathematics (I)
November 10, 2014: d:course:course110.tex
3
1.(20%) a, b, c, d ∈ R . By using the ε-δ identity. please prove
(a)(10%) (a x b)．(c x d) = (a．c)(b．d) - (a．d)(b．c),
(b)(10%) a x [b x (c x d)] = (b．d)(a x c) - (b．c)(a x d).
Hint: a．b = (a_k)(b_k), and (a x b)_i = (ε_ijk)(a_j)(b_k).
2.(30%)(a)(10%) For the following vectors:
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 1 │ │ 1 │ │ 2 │ │ 2 │
│ │ │ │ │ │ │ │
│ 1 │ │-1 │ │ 1 │ │-1 │
│ │, │ │, │ │, │ │,
│ 0 │ │ 1 │ │ 1 │ │ 0 │
│ │ │ │ │ │ │ │
│ 1 │ │ 1 │ │ 3 │ │ 3 │
└ ┘ └ ┘ └ ┘ └ ┘
are they linearly independent (LI) or linearly dependent? It means that you
need to solve
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 1 │ │ 1 │ │ 2 │ │ 2 │ │ 0 │
│ │ │ │ │ │ │ │ │ │
│ 1 │ │-1 │ │ 1 │ │-1 │ │ 0 │
x │ │+ x │ │+ x │ │+ x │ │ = │ │,
1│ 0 │ 2│ 1 │ 3│ 1 │ 4│ 0 │ │ 0 │
│ │ │ │ │ │ │ │ │ │
│ 1 │ │ 1 │ │ 3 │ │ 3 │ │ 0 │
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘
and check that are all coefficient x_1, x_2, x_3, x_4 zero or not.
(b)(10%) Please write the coefficient matrix A, with Ax = 0 representing the
above equation, and use the Gauss elimination method to find the solution of
x = (x_1, x_2, x_3, x_4)^T.
(c)(10%) Compute rank(A), nullity(A), the LI bases of rpw space of A, and the
LI bases of column space of A.
3.(20%)(a)(10%) Consider vectors
┌ ┐ ┌ ┐
│ 4 │ │ x │
│ │ │ 1│
a = │-1 │, x = │ x │,
│ │ │ 2│
│ 2 │ │ x │
└ ┘ └ 3┘
and derive a projection matrix P (in terms of a and its transpose a^T), such
that Px is the projection of x onto a. Hint: a．x = a^Tx,
(b)(10%) Let
┌ ┐
│ 2 │
│ │
x = │-1 │.
│ │
│ 3 │
└ ┘
Find the vector component of x along a and the vector component of x
orthogonol to a.
4.(30%) For the following matrices:
┌ ┐
┌ ┐ ┌ ┐ │ 1 4 5 0 9 │
│ 1 -1 3 │ │ 1 0 1 1 │ │ │
│ │ │ │ │ 3 -2 1 0 -1 │
A = │ 5 6 4 │, A = │ 3 2 5 1 │, A = │ │,
1 │ │ 2 │ │ 3 │-1 0 -1 0 -1 │
│ 7 4 2 │ │ 0 4 4 -4 │ │ │
└ ┘ └ ┘ │ 2 3 5 1 8 │
└ ┘
finding (a) a basis for the range of A_1, A_2, A_3 (column space), (b) a basis
for the kernal of A_1, A_2, A_3 (null space), (c) the rank and nullity of
A_1, A_2, A_3.
5.(20%) For the following matrices:
┌ ┐
┌ ┐ │ 2 0 -4 6 │
│-1 1 2 │ │ │
│ │ │ 4 5 1 0 │
A = │ 3 -1 1 │, A = │ │,
1 │ │ 2 │ 0 2 6 -1 │
│-1 3 4 │ │ │
└ ┘ │-3 8 9 1 │
└ ┘
-1
finding det(A) and A .