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成功大學 114 年度 線性代數

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Single-choice questions

第 1 題4 分
Let A=[1200013000145001]A = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 4 \\ 5 & 0 & 0 & 1 \end{bmatrix}. Which of the following statements is correct? (A) det⁡(A)=−119\det(A) = -119 (B) det⁡(A)=121\det(A) = 121 (C) det⁡(A)=−121\det(A) = -121 (D) det⁡(A)=119\det(A) = 119
第 2 題4 分
Find aa to ff so that the matrix AA is skew-symmetric. A=[0−2a0b0c82−70d0e60]A = \begin{bmatrix} 0 & -2 & a & 0 \\ b & 0 & c & 8 \\ 2 & -7 & 0 & d \\ 0 & e & 6 & 0 \end{bmatrix}. (A) a=−8,b=−6,c=0,d=2,e=−2a = -8, b = -6, c = 0, d = 2, e = -2 (B) a=−2,b=2,c=7,d=−6,e=−8a = -2, b = 2, c = 7, d = -6, e = -8 (C) a=8,b=6,c=0,d=−2,e=2a = 8, b = 6, c = 0, d = -2, e = 2 (D) a=2,b=−2,c=−7,d=6,e=8a = 2, b = -2, c = -7, d = 6, e = 8
第 3 題4 分
Which of the following matrices can be factorized as A=LUA = LU, where LL is a lower triangular matrix and UU is an upper triangular matrix? (A) [021134210]\begin{bmatrix} 0 & 2 & 1 \\ 1 & 3 & 4 \\ 2 & 1 & 0 \end{bmatrix} (B) [121242122]\begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 2 \end{bmatrix} (C) [221223405]\begin{bmatrix} 2 & 2 & 1 \\ 2 & 2 & 3 \\ 4 & 0 & 5 \end{bmatrix} (D) [21001−1−211]\begin{bmatrix} 2 & 1 & 0 \\ 0 & 1 & -1 \\ -2 & 1 & 1 \end{bmatrix}
第 4 題4 分
Let X=[x11x12x21x22x31x32]X = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \end{bmatrix} be a real matrix such that [−10111031−1]X=[12−311−1]\begin{bmatrix} -1 & 0 & 1 \\ 1 & 1 & 0 \\ 3 & 1 & -1 \end{bmatrix} X = \begin{bmatrix} 1 & 2 \\ -3 & 1 \\ 1 & -1 \end{bmatrix}. Which of the following statements is correct? (A) x12=2x_{12} = 2 (B) x21=5x_{21} = 5 (C) x11=−1x_{11} = -1 (D) x32=2x_{32} = 2
第 5 題4 分
Let A=[−1320−2110−2]A = \begin{bmatrix} -1 & 3 & 2 \\ 0 & -2 & 1 \\ 1 & 0 & -2 \end{bmatrix} and let adj(A)=[c11c12c13c21c22c23c31c32c33]\text{adj}(A) = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} denote the classical adjoint of AA. Which of the following statements is correct? (A) c12=4c_{12} = 4 (B) c21=4c_{21} = 4 (C) c11=−1c_{11} = -1 (D) c32=3c_{32} = 3
第 6 題4 分
Let A=[1234014100120002][1000020000−213−1010−222−121−12][214264224284−1201][100062004330−1201]A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 4 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 2 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 200 & 0 & 0 \\ -213 & -10 & 1 & 0 \\ -222 & -12 & 1 & -12 \end{bmatrix} \begin{bmatrix} 2 & 1 & 4 & 2 \\ 6 & 4 & 2 & 2 \\ 4 & 2 & 8 & 4 \\ -1 & 2 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 6 & 2 & 0 & 0 \\ 4 & 3 & 3 & 0 \\ -1 & 2 & 0 & 1 \end{bmatrix}. Find the rank of AA. (A) rank(A)=1\text{rank}(A) = 1 (B) rank(A)=2\text{rank}(A) = 2 (C) rank(A)=3\text{rank}(A) = 3 (D) rank(A)=4\text{rank}(A) = 4
第 7 題4 分
Solve kk such that the following system $\begin{cases} x_1 + 2x_2 - 3x_3 = 4 \\ 3x_1 - x_2 + 5x_3 = 2 \\ 4x_1 + x_2 + (k^2 - 14)x_3 = k + 2 \end{cases}$ has infinitely many solutions. (A) k=2k = 2 (B) k=4k = 4 (C) k=−4k = -4 (D) k=8k = 8
第 8 題4 分
Let the subspace W⊆R3W \subseteq \mathbb{R}^3 be defined as: W=span{[1−12],[−231]}W = \text{span} \left\{ \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix} \right\}. Which of the following vectors lies in the orthogonal complement W⊥W^\perp? (A) [111]\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} (B) [110]\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} (C) [−111]\begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} (D) [−7−51]\begin{bmatrix} -7 \\ -5 \\ 1 \end{bmatrix}
第 9 題4 分
Let u=[−1,2]u = [-1, 2] and v=[3,−5]v = [3, -5] in R2\mathbb{R}^2, and let T:R2→R3T : \mathbb{R}^2 \to \mathbb{R}^3 be a linear transformation such that T(u)=[−2,1,0]T(u) = [-2, 1, 0] and T(v)=[5,−7,1]T(v) = [5, -7, 1]. Let A=[a11a12a21a22a31a32]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} be the standard matrix representation of TT. Which of the following is TRUE? (A) a11=0a_{11} = 0 (B) a12=1a_{12} = 1 (C) a21=−2a_{21} = -2 (D) a22=1a_{22} = 1
第 10 題4 分
Which of the following sets of functions is linearly independent? (A) {ex,e−x,ex+e−x}\{e^x, e^{-x}, e^x + e^{-x}\} (B) {ex,ex+1,cos⁡(x),sin⁡(x)}\{e^x, e^{x+1}, \cos(x), \sin(x)\} (C) {ex,e−x,cosh⁡(x),sinh⁡(x)}\{e^x, e^{-x}, \cosh(x), \sinh(x)\} (D) {ex,e2x,e3x}\{e^x, e^{2x}, e^{3x}\}
第 11 題4 分
T:R2×3→R2×2T : \mathbb{R}^{2\times 3} \to \mathbb{R}^{2\times 2} defined by T([a11a12a13a21a22a23])=[2a11−a12a13+a1200]T \begin{pmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 2a_{11} - a_{12} & a_{13} + a_{12} \\ 0 & 0 \end{bmatrix}. Find rank(T)\text{rank}(T). (A) rank(T)=1\text{rank}(T) = 1 (B) rank(T)=2\text{rank}(T) = 2 (C) rank(T)=3\text{rank}(T) = 3 (D) rank(T)=4\text{rank}(T) = 4
第 12 題4 分
Let A=[−211−3]A = \begin{bmatrix} -2 & 1 \\ 1 & -3 \end{bmatrix}, B=[1000]B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, C=[−2000]C = \begin{bmatrix} -2 & 0 \\ 0 & 0 \end{bmatrix}, D=[2−10−12−10−12]D = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}. Classify matrices AA, BB, CC, and DD according to the categories: (1) Positive Definite, (2) Negative Definite, (3) Positive Semi-Definite, (4) Negative Semi-Definite. Which of the following gives the correct order of matrices from category (1) to (4)? (A) BACD (B) ADCB (C) ADBC (D) DABC
第 13 題4 分
Which of the following statements is FALSE about the algebraic and geometric multiplicities of the eigenvalues of AA? A=[210−101131]A = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 1 \\ 1 & 3 & 1 \end{bmatrix}. (A) The algebraic multiplicity of the eigenvalue −1-1 is 1. (B) The algebraic multiplicity of the eigenvalue 2 is 2. (C) The geometric multiplicity of the eigenvalue −1-1 is 1. (D) The geometric multiplicity of the eigenvalue 2 is 2.
第 14 題4 分
W1={[abca]:a,b,c∈R}W_1 = \left\{ \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \mathbb{R} \right\}, W2={[0a−ab]:a,b∈R}W_2 = \left\{ \begin{bmatrix} 0 & a \\ -a & b \end{bmatrix} : a, b \in \mathbb{R} \right\}. Find the dimension of W1∩W2W_1 \cap W_2. (A) dim⁡(W1∩W2)=1\dim(W_1 \cap W_2) = 1 (B) dim⁡(W1∩W2)=2\dim(W_1 \cap W_2) = 2 (C) dim⁡(W1∩W2)=3\dim(W_1 \cap W_2) = 3 (D) dim⁡(W1∩W2)=4\dim(W_1 \cap W_2) = 4
第 15 題4 分
A=[2112]A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. Compute eAe^A. (A) eA=[e32e2e2e32]e^A = \begin{bmatrix} \frac{e^3}{2} & \frac{e}{2} \\ \frac{e}{2} & \frac{e^3}{2} \end{bmatrix} (B) eA=[e200e32]e^A = \begin{bmatrix} \frac{e}{2} & 0 \\ 0 & \frac{e^3}{2} \end{bmatrix} (C) eA=[e+e32−e+e32−e+e32e+e32]e^A = \begin{bmatrix} \frac{e+e^3}{2} & \frac{-e+e^3}{2} \\ \frac{-e+e^3}{2} & \frac{e+e^3}{2} \end{bmatrix} (D) eA=[e+e32−e+e32−e+e32e+e32]e^A = \begin{bmatrix} \frac{e+e^3}{2} & \frac{-e+e^3}{2} \\ \frac{-e+e^3}{2} & \frac{e+e^3}{2} \end{bmatrix}
第 16 題4 分
Let T:R2→R2T : \mathbb{R}^2 \to \mathbb{R}^2 be a function. Which of the following defines a linear transformation? (A) T(a1,a2)=(1,a1)T(a_1, a_2) = (1, a_1) (B) T(a1,a2)=(a1,a22)T(a_1, a_2) = (a_1, a_2^2) (C) T(a1,a2)=(a1+1,a2)T(a_1, a_2) = (a_1 + 1, a_2) (D) T(a1,a2)=(a1cos⁡θ−a2sin⁡θ,a1sin⁡θ+a2cos⁡θ)T(a_1, a_2) = (a_1 \cos \theta - a_2 \sin \theta, a_1 \sin \theta + a_2 \cos \theta) for a fixed angle θ\theta
第 17 題4 分
Which of the following statements is FALSE? (A) SS is a basis for a vector space VV if and only if SS is a minimal spanning set. (B) VV is a finite-dimensional vector space (dim⁡(V)<∞\dim(V) < \infty), and WW is a subspace of VV. If dim⁡(V)=dim⁡(W)\dim(V) = \dim(W), then W=VW = V. (C) Let VV and V′V' be vector spaces having the same finite dimension, and let T:V→V′T : V \to V' be a linear transformation. Then TT is one-to-one if and only if range(T)=V′\text{range}(T) = V'. (D) Let VV and V′V' be vector spaces of dimensions nn and mm, respectively. A linear transformation T:V→V′T : V \to V' is invertible if and only if m=nm = n.

Multiple-choice questions

第 18 題4 分
Which of the following matrices are unitary? (A) 12[11i−i]\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ i & -i \end{bmatrix} (B) [1i0i]\begin{bmatrix} 1 & i \\ 0 & i \end{bmatrix} (C) 13[2−2+i2+i2]\frac{1}{3} \begin{bmatrix} 2 & -2 + i \\ 2 + i & 2 \end{bmatrix} (D) [0111]\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}
第 19 題4 分
Considering the following matrix: A=[100−84−6819]A = \begin{bmatrix} 1 & 0 & 0 \\ -8 & 4 & -6 \\ 8 & 1 & 9 \end{bmatrix}. The eigenvalues of the matrix AA are (A) -1 (B) 1 (C) 6 (D) 7 (E) 2
第 20 題4 分
Which of the following statements are correct? (A) A linear system with fewer equations than unknowns may have no solution. (B) Every linear system with the same number of equations as unknowns has a unique solution. (C) A linear system with coefficient matrix AA has an infinite number of solutions if and only if AA can be row-reduced to an echelon matrix that includes some column containing no pivot. (D) If [A∣b][A|b] and [B∣c][B|c] are row-equivalent partitioned matrices, the linear systems Ax=bAx = b and Bx=cBx = c have the same solution set. (E) A linear system with a square coefficient matrix AA has a unique solution if and only if AA is row equivalent to the identity matrix.
第 21 題4 分
Which of the following statements are NOT correct? (A) Let AA be a real n×nn \times n matrix. If A2A^2 is invertible, then ATA^T and A3A^3 are invertible. (B) If AA and BB are invertible, then so is A+BA + B, and (A+B)−1=A−1+B−1(A + B)^{-1} = A^{-1} + B^{-1}. (C) Let AA be a real m×nm \times n matrix and BB a real n×mn \times m matrix. Then trace(AB)=trace(BA)\text{trace}(AB) = \text{trace}(BA). (D) Let AA be a real m×nm \times n matrix and BB a real n×mn \times m matrix. Then det⁡(AB)=det⁡(BA)\det(AB) = \det(BA). (E) Let AA and BB be real n×nn \times n matrices. Then AB=BAAB = BA.
第 22 題4 分
Which of the following statements are NOT correct? (A) Any linear operator on an nn-dimensional vector space that has fewer than nn distinct eigenvalues is not diagonalizable. (B) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly independent. (C) If λ\lambda is an eigenvalue of a linear operator TT, then each vector in the eigenspace EλE_\lambda is an eigenvector of TT. (D) A linear operator TT on a finite-dimensional vector space is diagonalizable if and only if the multiplicity of each eigenvalue λ\lambda equals the dimension of the corresponding eigenspace EλE_\lambda. (E) If AA is diagonalizable, then ATA^T is also diagonalizable.
第 23 題4 分
Which of the following statements are NOT correct? (A) If SS is linearly independent and generates VV, each vector in VV can be expressed uniquely as a linear combination of vectors in SS. (B) Every vector space has at least two distinct subspaces. (C) No vector is its own additive inverse. (D) All vector spaces having a basis are finitely generated. (E) Any two bases in a finite-dimensional vector space VV have the same number of elements.
第 24 題4 分
Let W1W_1 and W2W_2 be subspaces of a finite-dimensional vector space VV. Let ⊕\oplus denote the direct sum. Which of the following statements are correct? (A) W1∩W2W_1 \cap W_2 is a subspace of VV. (B) W1∪W2W_1 \cup W_2 is a subspace of VV. (C) W1+W2W_1 + W_2 is a subspace of VV. (D) If V=W1⊕W2V = W_1 \oplus W_2, and β1\beta_1 and β2\beta_2 are bases for W1W_1 and W2W_2, respectively, then β1∩β2=∅\beta_1 \cap \beta_2 = \emptyset, and β1∪β2\beta_1 \cup \beta_2 is a basis for VV. (E) If W1⊕W2=VW_1 \oplus W_2 = V, then the dimension dim⁡(V)=dim⁡(W1)+dim⁡(W2)\dim(V) = \dim(W_1) + \dim(W_2).
第 25 題4 分
Which of the following statements are correct? (A) If QQ is orthogonal, then det⁡(Q)=±1\det(Q) = \pm 1. (B) Let AA be a real n×nn \times n matrix. Then AA is symmetric if and only if AA is orthogonally equivalent to a real diagonal matrix. (C) Let A∈Rn×nA \in \mathbb{R}^{n \times n} be a matrix whose characteristic polynomial splits over R\mathbb{R}. Then AA is orthogonally equivalent to a real upper triangular matrix. (D) Let TT be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space VV. Then every eigenvalue of TT is positive. (E) Let TT be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space VV. Then every eigenvalue of TT is negative.
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