课程名称︰线性代数
课程性质︰工管系科管组必修
课程教师︰苏柏青
开课学院：管理学院
开课系所︰工管系科管组
考试日期（年月日）︰2014.01.10
考试时限（分钟）：应该也是三节课
试题 :
1. (45%) Let
╭ 2 1 2 ╮
│ 0 3 4 │
│ 0 4 0 │
C = [c1 c2 c3] = │ 0 0 0 │
╰ 0 0 0 ╯
and let W = Col C = span {c1, c2, c3}
5
be the subspace of R for which the columns of C form a basis.
(a) (10%) Perform Gram-Schmidt Process on {c1, c2, c3} and find an
orthonormal basis {u1, u2, u3} for W (so that
span {c1}= span {u1} and span {c1, c2}= span {u1, u2}).
(If you get an orthogonal basis but not an orthonormal one,
you'll get 7 points.)
(b) (10%) Changing the order of the columns, perform Gram-Schmidt
Process on {c1, c3, c2} and find an orthonormal basis
{v1, v2, v3} for W (in this case, we want span {c1} = span {v1}
and span {c1, c3} = span {v1, v2}).
⊥
(c) (10%) Find an orthonormal basis for W , the orthogonal complement of
W. What is the dimension of W⊥?
T
(d) (5%) Let x= [1 2 3 4 5] . Try to decompose x into x = y + z so that
y ∈ W and Z ∈ W⊥.
(e) (10%) Find Pw, the 5 x 5 orthogonal projection matrix for W.
T -1 T
(Hint: Theorem 6.8 says Pw = C(C C) C, but this is not the only
way you can find the answer. Note that W is not only
span {c1, c2, c3}, but also span {u1, u2, u3} or
span {v1, v2, v3}.)
2. (25%) Let M be the set of all real n x n matrices. In the class we
n x n
have shown that M is a vector space.
n x n
(a) (10%) Let S be the set of all symmetric matrices, i.e.,
n x n
T
S = {A∈M : A = A }.
n x n n x n
Show that S is a subspace of M .
n x n n x n
(Hint: A subset of a vector space is a subplace thereof if and only
if it satisfies three conditions we discussed in class.)
(b) (10%) Let L (M , M ) be the set of all linear transformation
n x n m x m
from M to M . Then (M , M ) is also a
n x n m x m n x n m x m
vector space. What is the dimension of L (M , M )?
n x n m x m
(No derivation is required.)
(c) (5%) Let P3 be the set of all polynomials whose degrees are equal or
2 2 3
less than 3. We know that B = {1, 1+x, 1+x+x, 1+x+x+x } is also
3
a basis for P3. Express p(x) = x + x as a linear combination of
elements in B.
2 -1
3. (30%) Let A = [ ].
0 1
(a) (5%) Find the characteristic polynomial for A.
(b) (2%) Find all eigenvalues of A.
(c) (8%) For each of the eigenvalues of A, find the eigenspace corresponding
to the eigenvalue.
T
(d) (15%) Repeat (a)-(c) for A.
考古题可以放三年，觉得自己也太会拖
好久没打矩阵超麻烦，希望板主可以给些奖励金 >~<