7. Solve k such that the following system

 has infinitely many solutions.
(A)k=2
(B)k=4
(C)k=-4
(D)k=8

8. (4 points) Let the subspace be defined as:  Which of the following vectors les in the orthogonal complement ment (A)(B)(C)(D)

9.  Let u = [-1,2] and u = [3,-5] in and let T : be a linear transformation such that T(u) = [-2,1,0] and T(v)= (5,-7,1]. Let  be the standard matrix representation of T. Which of the following is TRUE? (A)(B)(C)(D)

10.  Which of the following sets of functions is linearly independent? (A) (B) (C)(D)

11.  T : defined by Find rank(T). (A) rank(T)=1(B) rank(T)=2 2 (C) rank(T)=3 (D) rank(T) = 4

12. Let  Classify matrices A, B, C, and D according to the categories: (1) Positive Definite, (2) Negative Definite, (3) Positive Semi-Defnite, (4) Negative Semi-Defite. Which of the following gives the correct order of matrices from category (1) to (4)? (A) BACD (B) ADCB (C) ADBC (D) DABC

13. Which of the following statements is FALSE about the algebraic and geomet- ric multiplicities of the eigenvalues of A? (A) The algebraic multiplicity of the eigenvalue -1 is 1. (B) The algebraic multiplicity of the eigenvalue 2 is 2. (C) The geometric multiplicity of the eigenvalue -1 is 1. (D) The geometric multiplicity of the eigenvalue 2 is 2.

14. ,, Find the dimen-sione of W1∩W2 sion of WinWz. (A) dim(Wi∩Wz)=1 (B) dim(W1∩W2)=2 (C) dim(W1∩W2)=3 (D) dim(W1∩W2)=4

15.  . (A)(B)(C)(D)

16. Let T : be a function. Which of the following defines a linear transformation? (A) T(a1, a2) = (1, a1) (B) T(a1, a2) = (a1, a2) (C) T(a1,a2) = (a1 + 1, a2) (D) T(a1,a2) = (a1 cosθ-a2sinθ, a1 sinθ+a2cos θ),   for a fixed angle θ

17.  Which of the following statements is FALSE? (A) S is a basis for a vector space V if and only if S is a minimal spanning set.  (B) V is a finite-dimensional vector space (dim(V) ＜∞), and W is a subspace of V. If dim(V) = dim(W), then W = V. (C) Let V and V'be vector spaces having the same finite dimension, and let T:V→ V be a linear transformation. Then T is one-to-one if and only if range(T) = V'  (D) Let V and V' be vector spaces of dimensions n and m, respectively. A linear transformation T : V -+ is invertible if and only if m = n.  

114年 - [無官方正解]114 臺灣綜合大學系統_學士班轉學生考試試題:線性代數#137898

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