大学MOOC Linear Algebra and Analytic Geometry(Harbin Institute of Technology)1461239169 最新慕课完整章节测试答案
Chapter Five Eigenvalues and Eigenvectors
文章目录
Quiz for Sec. 5.1
1、多选题：
Find the eigenvalues of 
.
选项：
A: 2
B: 1
C: 5
D: 0
答案: 【 2;
1;
5】
2、判断题：
If A is an 
matrix, and there exists an vector x such tha 
, then x is an eigenvector of A.
选项：
A: 正确
B: 错误
答案: 【 错误】
3、判断题：
A is an matrix. A is not invertible if and only if 0 is an eigenvalue of A.
选项：
A: 正确
B: 错误
答案: 【 正确】
4、判断题：
A is an matrix. Number c is an eigenvalue of A if and only if (A-cI)x=0 has nontrial solutions.
选项：
A: 正确
B: 错误
答案: 【 正确】
5、判断题：
A is an matrix. If 
are two linearly independent eigenvectors, then they must belongs to different eigenvalues.
选项：
A: 正确
B: 错误
答案: 【 错误】
Quiz for Sec. 5.2
1、判断题：
A is an matrix. Useing row replacement to A will not change the eigenvalue of A.
选项：
A: 正确
B: 错误
答案: 【 错误】
2、判断题：
A is an matrix. If c is a root of the characteristic equation with multiplicity 3, then the algebraic multiplity of c is 3.
选项：
A: 正确
B: 错误
答案: 【 正确】
3、判断题：
A and B are both matrices. A is similar to B then A and B have the same eigenvalues.
选项：
A: 正确
B: 错误
答案: 【 正确】
4、判断题：
A and B are both matrices. A and B have the same eigenvalues then A is similar to B.
选项：
A: 正确
B: 错误
答案: 【 错误】
5、判断题：
have the same eigenvalues.
选项：
A: 正确
B: 错误
答案: 【 正确】
Quiz for Sec. 5.3
1、判断题：
Suppose A, D, P are 
matrices. P is invertible, and 
, then A is diagonalizable.
选项：
A: 正确
B: 错误
答案: 【 错误】
2、判断题：
Suppose A is an matrix. 
has a bsis formed by the eigenvectors of A. The matrix A is diagonalizable.
选项：
A: 正确
B: 错误
答案: 【 正确】
3、判断题：
Suppose A is an matrix. A has n different eigenvectors then A is diagonalizable.
选项：
A: 正确
B: 错误
答案: 【 错误】
4、判断题：
Suppose A is an matrix. A has n different eigenvalues then A is diagonalizable.
选项：
A: 正确
B: 错误
答案: 【 正确】
5、判断题：
Suppose A is an matrix. A is diagonalizable, then A has n different eigenvalues.
选项：
A: 正确
B: 错误
答案: 【 错误】
Quiz for Sec. 5.4
1、判断题：
If matrix A is similar to matrix B then B is similar to A.
选项：
A: 正确
B: 错误
答案: 【 正确】
2、判断题：
If matrix A is similar to matrix B, B is similar to matrix C, then A is similar to C.
选项：
A: 正确
B: 错误
答案: 【 正确】
3、判断题：
If T is a linear transformation from an n-dimentional vector space V with a basis 
to an m-dimentional vector space W with basis 
, then the T matrix relative to basis and is an n by m matrix.
选项：
A: 正确
B: 错误
答案: 【 错误】
4、判断题：
If T is a linear transformation from an n-dimentional vector space V with a basis and , then the T matrix relative to basis is simlar to the T matrix relative to basis .
选项：
A: 正确
B: 错误
答案: 【 正确】
5、判断题：
A is an 
diagonalizable matrix. 
is a transformation from 
to . There exists a proper basis in , such that the T matrix relative to basis is a diagonal matrix.
选项：
A: 正确
B: 错误
答案: 【 正确】
Test for Chapter 5
1、单选题：
Which of the flowing vectors is an eigenvector of 
?
选项：
A: 
B: 
C: 
D: 
答案: 【 】
2、单选题：
Which one of the following numbers is not an eigenvalue of A?
选项：
A: 4
B: 1
C: 2
D: 3
答案: 【 4】
3、单选题：
Which of the following matrix may be not diagonalizable?
选项：
A: A is a 
matrix with two different eigenvalues. One eigenspace is three-dimensional and the other eigenspace is two-dimennsional.
B: A is 
matrix with three eigenvalues. One eigenspace is two-dimensional.
C: A is a 
matrix with three eigenvalues. One eigenspace is two-dimensional, and one of the other eigenspaces is three-dimensional.
D: A is a 
matrix with 3 different eigenvalues.
答案: 【 A is a matrix with three eigenvalues. One eigenspace is two-dimensional, and one of the other eigenspaces is three-dimensional.】
4、单选题：
Matrix 
is diagonalizable. There exist diagonal matrix D and invertible matrix P such that 
, 
. Which of the following matrix is P?
选项：
A: 
B: 
C: 
D: 
答案: 【 】
5、单选题：
Let 
and 
be bases for vector spaces V and W, repectively. Let 
be a linear transformation with the property that
Which is the matrix for T relative to 
and 
?
选项：
A: 
B: 
C: 
D: 
答案: 【 】
6、单选题：
Which of the following matrix is the 
-matrix of the transformation 
, when 
.
, 
选项：
A: 
B: 
C: 
D: 
答案: 【 】
7、单选题：
Define 
by 
Which of the following bases 
for 
with the property that 
is not diagonal.
.
选项：
A: 
B: 
C: 
D: 
答案: 【 】
8、多选题：
Choose the true statements. There are more than one right choices.
选项：
A: If A is invertible and 1 is an eigenvlaue for A, then 1 is an eigenvalue for 
B: If A contains a row or column of zeros, then 0 is an eigenvalue of A.
C: Eigenvalues must be nonzero scalars.
D: Eigenvectors must be nonzero vectors.
E: Two distinct eigenvectors are linearly independent.
F: Two eigenvectors corresponding to the same eigenvalue are always linearly dependent,
G: The sum of two eigenvectors of a matrix A is also an eigenvector of A.
H: The eigenvalues of an upper triangular matrix A are the nonzero entries on the diagonal of A.
答案: 【 If A is invertible and 1 is an eigenvlaue for A, then 1 is an eigenvalue for ;
If A contains a row or column of zeros, then 0 is an eigenvalue of A.;
Eigenvectors must be nonzero vectors.】
9、多选题：
Choose the false statements. There are more than one right choices.
选项：
A: If A is row equivalent to the identity matrix I, then A is diagonalizable.
B: Similar matrices always have exactly the same eigenvalues.
C: Similar matrices always have exactly the same eigenvectors.
D: If A is diagonalizable, then the columns of A are linearly independent.
E: If an 
matrix A has n different eigenvectors, then A is diagonalizable.
F: An matrix A is diagonalizable if A has n different eigenvalues.
G: If the dimention of the eigenspace of every eigenvalue of A equals the algebraic multiplicy of the eigenvalue, then A is diagonalizable.
答案: 【 If A is row equivalent to the identity matrix I, then A is diagonalizable.;
Similar matrices always have exactly the same eigenvectors.;
If A is diagonalizable, then the columns of A are linearly independent.;
If an matrix A has n different eigenvectors, then A is diagonalizable.】
Chapter Four Vector Spaces
Quiz for Sec. 4.2
1、多选题：
The matrix A has m rows and n columns. T(x)=Ax is a linear trasformation from 
to 
. Then which of the following statements is right.
选项：
A: T is one-to-one if the kernel of T only contains the zero vector.
B: T is onto if the range of T is .
C: If T is one-to-one and onto, then m=n.
D: The image of T is in .
答案: 【 T is one-to-one if the kernel of T only contains the zero vector.;
T is onto if the range of T is .;
If T is one-to-one and onto, then m=n.】
2、判断题：
The matrix A has m rows and n columns. The column space of A is a subspace of 
.
选项：
A: 正确
B: 错误
答案: 【 错误】
3、判断题：
The matrix A has m rows and n columns. The null space of A is a subspace of .
选项：
A: 正确
B: 错误
答案: 【 正确】
4、判断题：
The matrix A has m rows and n columns. A vector v is in Col A if Ax=v is consistent.
选项：
A: 正确
B: 错误
答案: 【 正确】
5、判断题：
The matrix A has m rows and n columns. A vector v is in Nul A if Av=0.
选项：
A: 正确
B: 错误
答案: 【 正确】
Quiz for Sec. 4.4
1、单选题：
is a basis for a vector space V. [s
. Which one is the vector x?
选项：
A: 
B: 
C: 
D: 
答案: 【 】
2、单选题：
If 
,
,then 
选项：
A: 
B: 
C: 
D: 
答案: 【 】
3、判断题：
The coordinates of a vector relative to a basis in a vector space is unique.
选项：
A: 正确
B: 错误
答案: 【 正确】
4、判断题：
The vector space 
and 
are isomorphic.
选项：
A: 正确
B: 错误
答案: 【 错误】
5、判断题：
If V is a subspace of 
, then the coordinate vector of any vector in V relative to a basis also in .