102年 - 102 國立高雄大學_博士班招生考試試題_應用數學系(一般生、在職生)：線性代數#147992

1 (10) Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Show that

$$n = \operatorname{rank}(T) + \operatorname{nullity}(T).$$

a. (10) Show that \( \operatorname{rank}(AB) \leqslant \operatorname{rank}(B) \) and \( \operatorname{rank}(AB) \leqslant \operatorname{rank}(A) \).

b. (10) Show that \( \operatorname{rank}(A^{\top}A) = \operatorname{rank}(A) \).

c. (5) If \( A \) is invertible then show that \( \operatorname{rank}(AB) = \operatorname{rank}(B) \).

a. (10) Show that the eigenvalues of \( A \) are real.

b. (10) If \( \lambda{1} \) and \( \lambda{2} \) are distinct eigenvalues of \( A \) associated eigenvectors \( x{1} \) and \( x{2} \), then \( x{1} \) and \( x{2} \) are orthogonal.

a. (5) Show that \( T \) is a linear transformation.

b. (10) Let \( A = \left[ \begin{array}{ll}3 & 2\\ 2 & 0 \end{array} \right] \) and \( B = I_{2} \). Find the eigenvalues and eigenvectors of \( T \).

a. (10) Find \( \min{x\in S}\| Ax\| 2 \) and \( \max{x\in S}\| Ax\| 2 \) , where \( S = \{x\in \mathbb{R}^2\mid \| x\| _2 = 1\} \)

b. (10) Find \( x0 \in \mathbb{R}^2 \) such that \( \|Ax0 - b\|2 = \min{x \in \mathbb{R}^2} \|Ax - b\|_2 \).

6 (10) Let $A, B$ and $D$ be real matrices. Suppose that $A = A^\top$, $D = D^\top$ and $A$ is invertible. Then show that $\left[ \begin{array}{cc} A & B \\ B^\top & D \end{array} \right]$ is positive if and only if $A$ and $D - B^\top A^{-1}B$ are positive definite.