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）︰2016/01/13
考试时限（分钟）：180
试题 :
Part A
╭ 1 2 ╮
Problem 1 (15 pts). Let A = │ │ and let V = M (|R) be a four-dimensional
╰ 2 1 ╯ 2
vector space over |R. Define the linear transformation T: V → V by
-1
T(B) = ABA .
(1) Find the dimension of the eigenspace with the eigenvalue 1.
(2) Show that T is diagonalizable.
4
Problem 2 (15 pts). Let V = |R and let
╭ 3 0 -1 0 ╮ ╭ 1 ╮
│ │ │ │
│ 5 8 1 -7 │ │ 0 │ 4
A = │ │ ∈ M (|R) and w := │ │ ∈ |R .
│ 5 -1 -2 1 │ 4 │ 3 │
│ │ │ │
╰ 4 3 0 -3 ╯ ╰ 1 ╯
Let T: V → V denote the linear transformation given by T(v) = Av.
2
(1) Find the dimension of the T-cyclic subspace |R[T]w := span {w, Tw, T w,
|R
...}.
(2) Use (1) to find an invertible matrix P ∈ M (|R) such that
4
╭ 0 0 0 -1 ╮
│ │
-1 │ 1 0 0 3 │
P AP = │ │.
│ 0 0 1 1 │
│ │
╰ 0 1 0 5 ╯
Problem 3 (25 pts). Let
╭ 2 -2 0 -2 ╮
│ │
│-3 -2 -3 -1 │
A = │ │ ∈ M (|R).
│ 0 3 2 3 │ 4
│ │
╰ 6 4 5 3 ╯
(1) Determine the characteristic polynomial of A.
(2) Find the Jordan form J of A.
-1
(3) Find an invertible P ∈ M (|R) such that P AP = J.
4
5
Problem 4 (15 pts). Let A ∈ M (€). Suppose that ch (x) = x (x-3) and that
n A
2 3
rank A = rank A . Determine all possible Jordan forms of A.
Part B
Problem 5 (15 pts). Let
╭ 3 2 b_1 b_2 ╮
│ │
│-1 1 b_3 b_4 │
A = │ │ ∈ M (|R).
│ 0 0 3 2 │ 4
│ │
╰ 0 0 -1 1 ╯
Show that there exists an invertible P ∈ M (|R) such that
4
╭ 3 2 0 0 ╮
│ │
-1 │-1 1 0 0 │
P AP = │ │
│ 0 0 3 2 │
│ │
╰ 0 0 -1 1 ╯
if and only if b + b = 0 and b = 2b + 2b .
1 4 2 1 3
Problem 6 (15 pts). Let A, B ∈ M (€). Suppose that the eigenvalues of A, B are
n
2
all non-negative real numbers and that null(A) = null(A ) and null(B) =
2 2 2
null(B ). If A = B , prove that A = B.