Linear Algebra - 中国大学mooc

Chapter 1 Systems of Linear Equations

unit test 1

1、
    A、
    B、
    C、
    D、

2、In the following matrices, ( ) is not elementary matrices.
    A、
    B、
    C、
    D、

3、
    A、the elementary row operation of interchanging the first and second rows.
    B、the elementary row operation of interchanging the second and third rows.
    C、the elementary column operation of interchang- ing the first and second columns.
    D、the elementary column operation of interchang- ing the second and third columns.

4、If A is nonsingular, then A can be factored into a product of elementary matrices.

5、If A is a 3×3 matrix and a1 +2a2 −a3 = 0, then A must be singular.

6、If A and B are n×n matrices, then (A−B)2 =A2−2AB+B2.

7、Use back substitution to solve the following systems of equations:

8、

9、

10、

Homework for chapter 1

1、See PDF file for details.

Chapter 2 Determinants

unit test 2

1、
    A、1.
    B、6.
    C、-6.
    D、0.

2、Let A be a n×n matrix, then ( ).
    A、AAT =ATA.
    B、det(AAT ) = det(AT A).
    C、if det(A) = 0, then A is invertible.
    D、if det(A) = 0, then AT is invertible.

3、
    A、
    B、
    C、
    D、

4、det(A+B)=det(A)+det(B).

5、A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.

6、If A and B are row equivalent matrices, then their determinants are equal.

7、

8、

9、Let A and B be 3×3 matrices with det(A) = 4 and det(B) = 5. Find the value of det (A−1 B) = .

10、

Homework for chapter 2

1、See PDF file for details.

Chapter 3 Vector Space

unit test 3

1、
    A、1,1.
    B、2,2.
    C、2,1.
    D、1,2.

2、Let A be an m×n matrix, then ( ).
    A、AAT =ATA.
    B、if rank(A) = m, then AT A is invertible.
    C、if rank(A) = n, then AAT is invertible.
    D、if rank(A) = n, then AT A is invertible.

3、Let A be an 10 × 6 matrix, and rank(A) = 3, then we have ( ).
    A、rank(AT ) = 6.
    B、dim(N(AT )) = 6.
    C、dim(N(A)) = 3.
    D、dim(N(AT )) = 3.

4、If x1, x2, ..., xn span Rn, then they are linearly independent.

5、If A is an m × n matrix, then A and AT have the same rank.

6、If U is the reduced row echelon form of A, then A and U have the same row space.

7、Determine whether the set {(x1, x2)T |x1 + x2 = 0} form subpace of R2 (Yes/No)

8、

9、

10、

Homework for chapter 3

1、See PDF file for details.

Chapter 4

unit test 4

1、Let A be a n×n matrix (n ≥ 3) and A∗ is adjoint of A. Suppose that k 0,±1, then (kA)∗ = ( ).
    A、kA∗.
    B、kn−1A∗.
    C、knA∗.
    D、k−1A∗.

2、
    A、a1a2a3a4 − b1b2b3b4.
    B、a1a2a3a4 + b1b2b3b4.
    C、(a1a2−b1b2)(a3a4−b3b4).
    D、(a2a3−b2b3)(a1a4−b1b4).

3、
    A、λ = −2 and B = 0.
    B、λ = −2 and B 0.
    C、λ = 1 and B = 0.
    D、λ = 1 and B 0.

4、Let L : Rn → Rn be a linear transformation. If L(x1) = L(x2), then the vectors x1 and x2 must be equal.

5、If L : V → V is a linear transformation and x∈ker(L), then L(v+x)=L(v) for all v∈V.

6、Let A, B, and C be n×n matrices. If A is similar to B and B is similar to C, then A is similar to C.

7、

8、

9、For the linear transformations L((x1, x2, x3)T ) =(x1 + x2, 0)T mapping R3 into R2, find a matrix A = such that L(x) = Ax for every x in R3

10、

Homework for chapter 4

1、See PDF file for details.

Chapter 5

unit test 5

1、
    A、
    B、
    C、
    D、

2、
    A、
    B、
    C、
    D、

3、
    A、
    B、
    C、
    D、

4、If x and y are nonzero vectors in Rn, then the vector projection of x onto y is equal to the vector projection of y onto x.

5、If U, V , and W are subspaces of R3 and if U ⊥ V and V⊥W, then U⊥W.

6、If A is an m×n matrix, then AAT and ATA have the same rank.

7、Find the angle between the vectors v = (2,−3)T and w = (3,2)T

8、Find the point on the line y = 2x+1 that is closest to the point (5, 2).

9、

Homework for chapter 5

1、See PDF file for details.

Chapter 6 Eigenvalues

unit test 6

1、Let A and B be two n×n orthogonal matrices, choose the incorrect one ( ) .
    A、A + B is a orthogonal matrix.
    B、AB is a orthogonal matrix.
    C、A−1 is a orthogonal matrix.
    D、B−1 is a orthogonal matrix.

2、Let A and B be two n × n matrices, they have the same eigenvalues and n linear independent eigenvalues, then ( ).
    A、A is similar to B.
    B、A B, but |A − B| = 0.
    C、A = B.
    D、A is not necessarily similar to B, but det(A)= det(B).

3、Choose a false statement ( ).
    A、The rank of an m×n matrix A is equal to the number of nonzero singular values of A, where singular values are counted according to multiplicity.
    B、If A is an n×n matrix, then A and AT have the same eigenvectors.
    C、If A is symmetric, then eA is symmetric positive definite.
    D、Let A be a nonsingular n × n matrix and let be an eigenvalue of A, then λ−1 is an eigenvalue of A−1.

4、If A is an n × n matrix whose eigenvalues are all nonzero, then A is nonsingular.

5、If A and B are similar matrices, then they have the same eigenvalues.

6、If A has eigenvalues of multiplicity greater than 1, then A must be defective.

7、

8、

9、Let A be a 2×2 matrix. If tr(A)=8 and det(A)=12, the eigenvalues of A is .

Homework for chapter 6

1、See PDF file for details.