98年 - 98 國立高雄大學_博士班招生考試試題_應用數學學系：線性代數#147999

科目：研究所、轉學考(插大)◆線性代數 | 年份：98年 | 選擇題數：0 | 申論題數：7

試卷資訊

所屬科目：研究所、轉學考(插大)◆線性代數

選擇題 (0)

申論題 (7)

1. (25%) Let n n T : be a linear transformation with R → R T x Ax v v ( ) = , where A is an n× n
matrix. Write down as many as possible equivalent statements of “ A is invertible”, and sketch
the proof.
2. (15%) Find the eigenvalues of, eigenvectors of, and diagonalize the matrix 

 


−
− = 4 3
3 4
A and
explain the meanings of these calculations.
3. (15%) Let T be a linear transformation and λ λ λn , , , 1 2 L be distinct eigenvalues of T . If
n v v v
v L v v
, , , 1 2 are eigenvectors of T such that λi corresponds to i v
v (1 ≤ i ≤ n ), Prove that
n v v v
v L v v
, , , 1 2 are linear independent.
4. (15%) Let 2 2 T : R → R be a linear transformation preserving lengths and angles. Prove that T
is either a rotation or a refection about a line.
5. (15%) Prove or disprove:
(1) If A, are two B n× n matrices of the same rank, then they must be similar.
(2) If A is an invertible symmetric matrix, then 2 A must be positive definite.
(3) If A, are square matrices of the same size such that B ABAB = 0 , then BABA = 0
6. (15%) Let J be the Jordan canonical form of a linear transformation T . Prove that J can be
written as the sum of a diagonal matrix and a nilpotent matrix, and these two matrices commute.