上海交通大学2014年线性代数英文考卷

adj (B ) 0 3. If a matrix A is similar to another matrix B , then (A) λI − A = λI − B ; (C) det(A) = det(B ); (B) A and B have the same eigenvectors; (D) both A and B are similar to a diagonal matrix.
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1
−1 −1 −1
.
−1 1 −1 −1 −1 −1 1 −1
−1 −1 −1
1
(8 points)
5
20. Let W = SP (S ) be a subspace, where
0 2 1 1 S = 1 , 0 , 1 , 1 , −2 −1 −3 −1
.
2
14. Let V be the vector space of all (2 × 2) real matrices, and W be the subspace defined by W = {A : A = Exhibit a basis for W . 15. If

3a a−b
a + 2b b
8. Given the vectors 3 4 −3 −4 −6 −8
x1 = −2 , x2 = 2 , x3 = 4 .
Then the dimension of the subspace Span{x1 , x2 , x3 } spanned by x1 , x2 , x3 is (A) 1 (B) 2 (C) 3 (D) 0.
EXAMINATION OF LINEAR ALGEBRA
Class: Number :
Name: Score:
1
( )
Choose the best answer (24 points)
1. If A is m × n and b is nonzero m column vector with m < n, then linear system Ax = b (A) has only one solution; (C) is inconsistent; (B) has infinite many solutions;
Find a subset T of S such that T is a basis for W and determine dim(W ). (8 points)
21. Let A be an n × n positive definite matrix. (1). Prove that A−1 is positive definite matrix. A A (2). Prove that B = is positive semidefinite matrix, but is not positive A A definite. (10 points)
( )
2
Fill in (24 points)
9. Let {1, 1 + x, x2 } be a basis for the set A which is of all real polynomials of degree 2 or less. Find the coordinate of x2 + 3x + 5 with respect with the above basis. 10. Find general solutions to the linear system (in vector form) x1 + x2 − x3 − x4 = 0. 11. Let a 3 × 3 matrix A have three eigenvalues 1, 2, 3. Then det(A + 2I ) = 12. Let A = [A1 , A2 , A3 ] be a 3 × 3 matrix and det(A) = 2. If B = [A1 + A2 , A2 − A3 , A3 + A2 ], then det (B ) = 1 13. If u = 1 and v =
4. Let A be an m × n matrix. If the column vectors of A are linearly independent, then which of the following statements is False. ( ) (A) rank (A) = n (B) the system Ax = b always has at least one solution. In 0 (C)The dimension of the row space of A is n, (D) There exists nonsingular matrices P and Q such that P AQ = .
1
5. Let α1 , α2 , α3 be linear independent. Then which of the following vectors are linear independent. (A) α1 + α2 , α2 + α3 , α3 − α1 . (B)α1 + α2 , α2 + α3 , α1 + 2α2 + α3 . (C)α1 + 2α2 , 2α2 + 3α3 , −α1 + 3α3 . (D)α1 + α2 + α3 , 2α1 − 3α2 + 2α3 , 3α1 + 5α2 − 5α3 . 6.Let A, B and C matrices satisfy ABC = I , where I is the identity matrix. Then which of the following is true (A) ACB = I ; 7.
(D) all of the above three is not true.
2. Let A and B be 3 × 3 inevitable matrices. Then the adjacent matrices adj (C ) of 0 A C= is equal to ( ) B 0 (A) adj (C ) = (B)adj (C ) = (C) adj (C ) = (D)adj (C ) = 0 |B |adj (B ) 0 −|B |adj (B ) 0 −|B |adj (A) 0 adj (A) ( ) |A|adj (A) 0 −|A|adj (A) 0 −|A|adj (B ) 0
}.
1 1 −1 1 1 a

A= 0 1 is singular, then a = .
2

2 1 0

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16. The geometric multiplicities for the eigenvalue 2 of A = 0 2 1 is equal to 0 0 2 .
3
18. Discuss the solutions of the following linear system in unknowns x1 , x2 , x3
x1 − ax2 − 2x3
1 2 3 5x − 5x − 4x 1 2 3
= −1 = = 2 1
x − x + ax
6
22. Let A = (aij ) be an n × n matrix. Denote by A = (A1 , · · · , An ) and I − A = (B1 , · · · , Bn ). If A2 = A, then (1). All eigenvalues of A are either 0 or 1. (2). SP {A1 , · · · , An , B1 , · · · , Bn } = Rn . (3).prove that A is diagonalizable. ( 10 points).
.
1

1 x matrix A corresponding to the eigenvalues λ1 = 2 and λ2 = 3, respectively. Then x =
−1 be two real eigenvectors of a 3 × 3 real symmetric

( )
(B) BCA = I ;
(C) BAC = I ;
(D) CBA = I .
103 100 204
det 199 200 395 = 301 300 600
(
)
(A) 1000
(B) -1000

(C) -2000

(D) 2000.

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Calculation and Show

17. Let
1 1
−1 1 −1
1 1

A= −1
−1 .
(1). Find a column vector α such that A = ααT (2). If there exists a 3 × 3 matrix B such that A + B = AB , find out B . ( 8 points)
Determine the conditions for a such that the system is consistent and inconsistent. Moreover, find all solutions for consistent system. (8 points)
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19. Find an orthonormal basis for R4 consisting of eigenvectors of the matrix