课程名称︰线性代数一
课程性质︰数学系大一必修
课程教师︰谢铭伦
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/11/11
考试时限（分钟）：130
试题 :
Notation: F denotes the set of complex numbers, n is a positive integer and I
n
is the identity matrix in M (F).
n
Part A
2 3
Problem 1 (10 pts). Let T: F → F be the linear transformation such that
T((1, 2)) = (1, 1, 1); T((2, 1)) = (1, -1, 3).
Let A = {(-5, 2), (-1, 1)} and B = {(2, 3, 1), (1, 2, 1), (1, 1, 1)} be bases of
2 3
F and F . Find the matrix representation [T] of T with respect to A and B.
A, B
4 3
Problem 2 (20 pts). Let T: F → F be the linear transformation defined by T(v)
= A・v, where
╭ 3 -1 1 1 ╮
│ │
A = │ 1 1 -1 3 │ ∈ M (F).
│ │ 3 ×4
╰ 2 0 0 2 ╯
(1) Find the rank and the nullity of T.
(2) Find bases of Ker T and Im T.
3
Problem 3 (10 pts). Let V be the subspace of F spanned by
a
(1, 2, a), (1, a, 2), (a, 1, 2).
Determine all possible values a ∈ F such that dim V = 2.
F a
Problem 4 (15 pts). Let
╭ -1 2 -2 ╮
│ │
A = │ 7 -3 1 │.
│ │
╰ 11 -8 6 ╯
-1
Find an invertible matrix P ∈ M (F) such that P AP is a diagonal matrix.
3
Problem 5 (10 pts). Let
╭ 1 0 1 ╮ ╭ 1 1 1 ╮
│ │ │ │
A = │ -1 -3 1 │; B = │ 1 2 1 │.
│ │ │ │
╰ 1 4 1 ╯ ╰ 1 4 3 ╯
Find all roots of the characteristic polynomial of AB.
Part B
Problem 6 (10 pts). Let V be a finite dimensional vector space over F with
dim V = m. Let v , v , ..., v be n vectors in V. Show that if n ≧ m + 2, then
F 1 2 n
n
there exists α , α , ..., α in F not all equal to zero such that Σ α v =
1 2 n i=1 i i
n
0 and Σ α = 0.
i=1 i
Problem 7 (10 pts). Let A ∈ M (F). If rank A + rank(I + A) = n, show that
n n
Tr(A) + rank A = 0.
Problem 8 (15 pts). Let A, B ∈ M (F). Let C = AB - BA. If CA = AC, prove that C
n
is NOT invertible.