尔雅Linear Algebra_1课后答案(学习通2023题目答案)

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尔雅Linear Algebra_1课后答案(学习通2023题目答案)

Chapter 1-Matrices and Systems of Equations

test1

1、尔雅Let be a system of n linear equations in n unknowns and suppose that and are both solutions and . The后答 system has______solutions.
A、infinitely many
B、案学two
C、习通three
D、题目finite

2、答案Let A and B be 10×10 matrices that are partitioned into submatrices as follows ,尔雅. If is a 6 × 5 matrix, and is a k × r matrix, in order to make the block multiplication of A times B possible, k is ______ .
A、6
B、后答5
C、案学11
D、习通1

3、题目Let A be a matrix,答案 then ________.
A、If Ax=0 has only trivial solution,尔雅 then Ax=b must have a unique solution.
B、If Ax=0 has a nontrivial solution,后答 then Ax=b must have infinitely many solutions.
C、If Ax=b has a unique solution,案学 then Ax=0 must have a unique solution.
D、If Ax=0 has a nontrivial solution, then must have a nontrivial solution.

4、Let AB be all square matrices. There must be _______
A、
B、AB=BA
C、
D、

5、Let AB be all square matrices, if , then ______.
A、AB=I or AB=-I
B、
C、AB=BA
D、A=B=I

6、If the row reduced echelon form of A involves free variables, then the system Ax=b will have infinitely many solutions.

7、Every homogeneous linear system is consistent.

8、An matrix A is nonsingular if and only if the reduced row echelon form of A is I.

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

10、If A and B are nonsingular matrices, then A+B is also nonsingular and .

11、If , then A must be equal to either I or -I.

12、If A and B are matrices, then .

13、If AB=O, then BA=O.

14、If AC=BC and C=0 (the zero matrix), then A=B.

15、If A is a matrix and , then A must be singular.

Chapter 2 - Determinants

test2

1、If A is a nonsingular matrix, show that is ________ and det()=_______.
A、nonsingular >0
B、nonsingular <0
C、singular >0
D、singular <0

2、Let A be a 4×4 matrix, the determinant of A is 1/3, and the adjoint matrix of A, then =________.
A、1
B、3
C、6
D、9

3、,.,then =_______.
A、
B、
C、
D、

4、has only a unique solution, satisfies
A、
B、
C、 or
D、 and

5、 are 4×1 matrices, let , and |A|=1,|B|=2,the determinant of |A+B|=______
A、1
B、3
C、6
D、9

6、det(AB) = det(BA)

7、det(A-B) = det(A) ? det(B)

8、det(cA) = cdet(A)

9、

10、 implies

11、

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

13、If x and y are two distincts vectors in , and Ax=Ay, then det(A)=0

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

15、 but (where denotes the zero matrix) for some positive integer k, then must be singular.

Chapter 4-Linear Transformations

Test 4

1、1. Determine which the following are not linear operators on
A、 is the operator de?ned by
B、 is the operator de?ned by
C、 is the operator de?ned by
D、 is the operator de?ned by

2、Determine which the following are not linear operators on
A、
B、
C、
D、

3、Let be the linear operator on de?ned by Span constitutes the the range of with defined in
A、
B、
C、
D、

4、If the matrices and are similar, which of the following statement is wrong,
A、 and are similar.
B、 and are similar.
C、 and can be regarded as the same linear transformation matrix under different basis.
D、

5、The following matrices, is similar to , where the matrices and are defined in
A、
B、
C、
D、

6、Let be a linear transformation. If , then the vectors and must be equal.

7、Any two matrices with the same trace are similar.

8、If , are both linear operators on a vector space V, then is also a linear operator on V, where is the mapping de?ned by for all .

9、If is a linear trans formation and then

10、If rotates each vector x in by 45 degrees in the counterclockwise direction and then re?ects the resulting vector about the y-axis, and if is a transformation that does the same two operations, but in the reverse order, then .

11、The set of all vectors x used in the homogeneous coordinate system forms a subspace of

12、Let A, B and C be matrices. If A is similar to B and B is similar to C, then A is similar to C.

13、Let . be a linear transformation, and let A be the standard matrix representation of L. If is de?ned by for all then is a linear transformation and its standard matrix representation is

14、Let be an ordered basis for . If , and have the same matrix representation with respect to E, then .

15、Let be a linear transformation. If A is the standard matrix representation of , then an matrix B will also be a matrix representation of L if and only if B is similar to A.