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

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 \( A = \begin{bmatrix} -3 & 4 \\ 4 & -3 \end{bmatrix} \)  and       explain      the meanings of these calculations.

3. (15%) Let T be a linear transformation and \( \lambda_1, \lambda_2, \dots, \lambda_n \) be      distinct      eigenvalues of T . If \( \vec{v}_1, \vec{v}_2, \dots, \vec{v}_n \),  are eigenvectors of T such that λ_i corresponds to  \( \vec{v}_i \) (1 ≤ i ≤ n ), Prove that \( \vec{v}_1, \vec{v}_2, \cdots, \vec{v}_n \) are linear independent.

4. (15%) Let  T : R² → R² be a linear transformation preserving lengths and angles. Prove that T is either a rotation or a refection about a line.

(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.