112 年 國立成功大學電機工程學系碩士班已組《線性代數》

1. (30 pts) Make each statement True or False and JUSTIFY each answer. (3 pts for each question. Correct answer for 2 pts and suitable justification for 1 pts.)

(a) A is an $n \times n$ matrix. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by $(-1)^r$, where $r$ is the number of row interchanges made during row reduction from A to U.

(b) If a set $\{v_1, \dots, v_p\}$ spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.

(c) B and C are bases for a vector space V. If $V = \mathbb{R}^2$, $B = \{b_1, b_2\}$, and $C = \{c_1, c_2\}$, then row reduction of $[c_1 \ c_2 \mid b_1 \ b_2]$ to $[I \mid P]$ produces a matrix P that satisfies $[x]_B = P[x]_C$ for all $x$ in V.

(d) A is an $n \times n$ matrix. A row replacement operation on A does not change the eigenvalues.

(e) A, B, C are $n \times n$ matrices. If B is similar to A and C is similar to A, then B is similar to C.

(f) The Gram-Schmidt process produces from a linearly independent set $\{x_1, \dots, x_p\}$ an orthogonal set $\{v_1, \dots, v_p\}$ with the property that for each $k$, the vectors $v_1, \dots, v_k$ span the same subspace as that spanned by $x_1, \dots, x_k$.

(g) A is an $m \times n$ matrix and $b$ is in $\mathbb{R}^m$. A least-squares solution of $Ax = b$ is a list of weights that, when applied to the columns of A, produces the orthogonal projection of $b$ onto Col A.

(h) If $B = PDP^T$, where $P^T = P^{-1}$ and D is a diagonal matrix, then B is a symmetric matrix.

(i) The maximum value of $Q(x) = 7x_1^2 + 3x_2^2 - 2x_1x_2$, subject to the constraint $x_1^2 + x_2^2 = 1$, is $5 + \sqrt{5}$.

(j) A is an $m \times n$ matrix with a singular value decomposition $A = U\Sigma V^T$, where U is an $m \times m$ orthogonal matrix, $\Sigma$ is an $m \times n$ "diagonal" matrix with r positive entries and no negative entries, and V is an $n \times n$ orthogonal matrix. If P is an orthogonal $m \times m$ matrix, then PA has the same singular values as A.

2. (20 pts) Let $P_3$ have the inner product given by evaluation at -3, -1, 1, and 3. Let $p_0(t) = 1$, $p_1(t) = t$, and $p_2(t) = t^2$.

(a) (10 pts) Compute the orthogonal projection of $p_2$ onto the subspace spanned by $p_0$ and $p_1$.

(b) (10 pts) Find a polynomial $q$ that is orthogonal to $p_0$ and $p_1$ such that $\{p_0, p_1, q\}$ is an orthogonal basis for span$\{p_0, p_1, p_2\}$. Scale the polynomial $q$ so that its vector of values at (-3, -1, 1, 3) is (1, -1, -1, 1).

3. (15 pts) Find a factorization $A = QR$.
   $A = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$

4. (35 pts, 7 pts each) Let A and its reduced echelon form be given as:
   $A = \begin{pmatrix} -1 & -5 & 3 & 9 \\ -48 & -40 & 24 & 92 \\ 94 & 70 & -42 & -166 \\ -48 & -40 & 24 & 92 \end{pmatrix}, \quad U = \begin{pmatrix} 1 & 0 & -3 & -17 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$
   We already know that $\lambda = 4$ is an eigenvalue of A, and $u = [1, 0, 2, 0]^T$ is an eigenvector of A.

(a) Find a basis for the eigenspace for $\lambda = 4$.

(b) What is the eigenvalue for the eigenvector u?

(c) Notice that the second and fourth rows of A are the same. Does that imply we have a certain eigenvalue? Find a basis for its eigenspace. To save you some time, we have included the reduced echelon form of A.

(d) What is the characteristic polynomial of A?

(e) Show that A is diagonalizable by finding an appropriate P and D.