97年 - 97 國立高雄大學_博士班招生考試試題_統計學博士班：線性代數#147983

科目：研究所、轉學考(插大)◆線性代數 | 年份：97年 | 選擇題數：0 | 申論題數：15

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所屬科目：研究所、轉學考(插大)◆線性代數

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申論題 (15)

1 (24) Determine "true" or "false" for the following statements. Briefly sketch your proof when the answer is "true", or give a counterexample when the answer is "false".

a. If \(A^2 = A\) then the eigenvalues of \(A\) are either 0 or 1.
b. Let \(A, B \in M_{n \times n}(\mathbb{R})\). If \(AB = 0\) then \(\operatorname{rank}(A) + \operatorname{rank}(B) \leq n\).
c. If \(A, B \in M_{n \times n}(\mathbb{R})\) then \(AB\) and \(BA\) have the same minimal polynomial.
d. If \(A, B \in M_{n \times n}(\mathbb{R})\) then the determinant of \(AB - BA\) is zero.
2 Let $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 1 & 2 & 4 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 1 \\ -1 \\ 5 \end{bmatrix}$.

a. (6) Find the projection of \(\mathbf{b}\) onto the column space of \(A\).
b. (6) Let \(\mathbf{b}_A\) be the projection of \(\mathbf{b}\) onto the column space of \(A\). Find solution \(\mathbf{s}\) to \(A\mathbf{x} = \mathbf{b}_A\) such that \(\| \mathbf{s} \| \leq \| \mathbf{u} \|\) for all other solutions \(\mathbf{u}\).
3 (10) Let \(A \in M_{n \times m}(\mathbb{R})\), \(B \in M_{m \times n}(\mathbb{R})\). Prove that if \(\lambda\) is an eigenvalue of \(AB\) and \(\lambda \neq 0\) then \(\lambda\) is an eigenvalue of \(BA\).
4 (10) Let \( A \in M_{n \times n}(\mathbb{R}) \) and \( r \in \mathbb{R} \). Suppose \( A - rI_n = QR \) where \( Q \) is orthogonal and \( R \) is upper-triangular. Let \( B = RQ + rI_n \). Show that \( A \) and \( B \) have the same eigenvalues.
5 Let \(\mathbf{x},\mathbf{y}\in \mathbb{R}^n\) and \(A = I_{n} - \mathbf{x}\mathbf{y}^{T}\)

a. (5) Let $\mathbf{x}^T\mathbf{y} \neq 1$. Show that

$$A^{-1} = I_n - \frac{1}{\mathbf{x}^T\mathbf{y} - 1}\mathbf{x}\mathbf{y}^T.$$
b. (8) Let \(\mathbf{x}^T\mathbf{y}\neq 0\) . Find the eigenvalues and corresponding to eigenspaces of \(A\)
c. (5) Show that \(\operatorname{det}(A) = 1 - \mathbf{x}^T\mathbf{y}\).
d. (8) Let \(\mathbf{x}\) and \(\mathbf{y}\) be nonzero vectors. Show that if \(\mathbf{x}^T\mathbf{y} = 0\) then \(A\) is not diagonalizable.
6 Let $A = \begin{bmatrix} 1 & 4 \\ 3 & 2 \end{bmatrix}$. Define $T : M_{2 \times 2}(\mathbb{R}) \to M_{2 \times 2}(\mathbb{R})$ such that $T(X) = XA - AX$.

a. (5) Show that \(T\) is a linear transformation.
b. (10) Find the eigenvalues and eigenvectors of \(T\).
c. (3) Find the minimal polynomial of \(T\).