Exercises

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Appendix B Exercises

0.1 Sets

0.2 Functions

0.4 Logic

Exercise 0.4.3.1

Exercise 0.4.3.2

Exercise 0.4.3.3

Exercise 0.4.3.4

Exercise 0.4.3.5

Exercise 0.4.3.6

0.6 Complex numbers

Exercise 0.6.3.1

Exercise 0.6.3.2

Exercise 0.6.3.3

Exercise 0.6.3.4

0.7 Polynomials

Exercise 0.7.4.1

Exercise 0.7.4.2

1.1 Systems of linear equations

Exercise 1.1.3.1

Exercise 1.1.3.2

Exercise 1.1.3.3

Exercise 1.1.3.4

Exercise 1.1.3.5

Exercise 1.1.3.6 Geometry of linear systems

Exercise 1.1.3.7 Row operations preserve solutions

Exercise 1.1.3.8 Nonlinear systems

Exercise 1.1.3.9 Not all arithmetic operations preserve solutions

1.2 Gaussian elimination

Exercise 1.2.3.1

Exercise 1.2.3.2

Exercise 1.2.3.3

Exercise 1.2.3.4

Exercise 1.2.3.5

Exercise 1.2.3.6

Exercise 1.2.3.7

Exercise 1.2.3.8

Exercise 1.2.3.9

Exercise 1.2.3.10

Exercise 1.2.3.11

1.3 Solving linear systems

Exercise 1.3.3.1

Exercise 1.3.3.2

Exercise 1.3.3.3

Exercise 1.3.3.4

Exercise 1.3.3.5

Exercise 1.3.3.6

Exercise 1.3.3.7

Exercise 1.3.3.8

Exercise 1.3.3.9

Exercise 1.3.3.10

Exercise 1.3.3.11

Exercise 1.3.3.12

Exercise 1.3.3.13

Exercise 1.3.3.14

Exercise 1.3.3.15

Exercise 1.3.3.16

Exercise 1.3.3.17

Exercise 1.3.3.18

2.1 Matrix arithmetic

Exercise 2.1.6.1

Exercise 2.1.6.2

Exercise 2.1.6.3

Exercise 2.1.6.4

Exercise 2.1.6.5

Exercise 2.1.6.6

Exercise 2.1.6.7

Exercise 2.1.6.8

Exercise 2.1.6.9

Exercise 2.1.6.10

Exercise 2.1.6.11

2.2 Matrix algebra

2.3 Invertible matrices

Exercise 2.3.3.1

Exercise 2.3.3.2

Exercise 2.3.3.3

Exercise 2.3.3.4

Exercise 2.3.3.5

Exercise 2.3.3.6

Exercise 2.3.3.7

Exercise 2.3.3.8

Exercise 2.3.3.9

Exercise 2.3.3.10

Exercise 2.3.3.11

Exercise 2.3.3.12

Exercise 2.3.3.13

Exercise 2.3.3.14 Expanding matrix products

Exercise 2.3.3.15 Polynomial expressions of

A

commute

Exercise 2.3.3.16

2.4 The invertibility theorem

Exercise 2.4.7.1

Exercise 2.4.7.2

Exercise 2.4.7.3

Exercise 2.4.7.4

Exercise 2.4.7.5

Exercise 2.4.7.6

Exercise 2.4.7.7

Exercise 2.4.7.8

Exercise 2.4.7.9

Exercise 2.4.7.10

Exercise 2.4.7.11

Exercise 2.4.7.12

Exercise 2.4.7.13

Exercise 2.4.7.14

Exercise 2.4.7.15

Exercise 2.4.7.16

Exercise 2.4.7.17

Exercise 2.4.7.18

Exercise 2.4.7.19

Exercise 2.4.7.20

Exercise 2.4.7.21

Exercise 2.4.7.22 Properties of row equivalence

Exercise 2.4.7.23

Exercise 2.4.7.24

Exercise 2.4.7.25

Exercise 2.4.7.26

2.5 The determinant

Exercise 2.5.4.1

Exercise 2.5.4.2

Exercise 2.5.4.3

Exercise 2.5.4.4

Exercise 2.5.4.5

Exercise 2.5.4.6

Exercise 2.5.4.7

Exercise 2.5.4.8

Exercise 2.5.4.9

Exercise 2.5.4.10

Exercise 2.5.4.11

Exercise 2.5.4.12

Exercise 2.5.4.13

Exercise 2.5.4.14

Exercise 2.5.4.15

Exercise 2.5.4.16

Exercise 2.5.4.17

Exercise 2.5.4.18

Exercise 2.5.4.19

Exercise 2.5.4.20

Exercise 2.5.4.21

Exercise 2.5.4.22

Exercise 2.5.4.23

Exercise 2.5.4.24

Exercise 2.5.4.25

Exercise 2.5.4.26

Exercise 2.5.4.27

Exercise 2.5.4.28

3.1 Real vector spaces

Exercise 3.1.4.1

Exercise 3.1.4.2

Exercise 3.1.4.3

Exercise 3.1.4.4

Exercise 3.1.4.5

Exercise 3.1.4.6

Exercise 3.1.4.7

Exercise 3.1.4.8

Exercise 3.1.4.9

Exercise 3.1.4.10

Exercise 3.1.4.11

Exercise 3.1.4.12

Exercise 3.1.4.13

Exercise 3.1.4.14

3.2 Linear transformations

Exercise 3.2.6.1

Exercise 3.2.6.2

Exercise 3.2.6.3

Exercise 3.2.6.4

Exercise 3.2.6.5

Exercise 3.2.6.6

Exercise 3.2.6.7

Exercise 3.2.6.8

Exercise 3.2.6.9 Transposition

Exercise 3.2.6.10 Scalar multiplication

Exercise 3.2.6.11 Trace

Exercise 3.2.6.12 Left/right matrix multiplication

Exercise 3.2.6.13 Conjugation

Exercise 3.2.6.14 Sequence shift operators

Exercise 3.2.6.15 Function shift operators

Exercise 3.2.6.16 Function scaling operators

Exercise 3.2.6.17 Adding and scaling linear transformations

Exercise 3.2.6.18

Exercise 3.2.6.19

Exercise 3.2.6.20 Reflection through a line

Exercise 3.2.6.21 Compositions of rotations and reflections

3.3 Subspaces

Exercise 3.3.5.1

Exercise 3.3.5.2

Exercise 3.3.5.3

Exercise 3.3.5.4

Exercise 3.3.5.5

Exercise 3.3.5.6

Exercise 3.3.5.7

Exercise 3.3.5.8

Exercise 3.3.5.9

Exercise 3.3.5.10

Exercise 3.3.5.11

Exercise 3.3.5.12

Exercise 3.3.5.13

Exercise 3.3.5.14

Exercise 3.3.5.15

Exercise 3.3.5.16

Exercise 3.3.5.17

Exercise 3.3.5.18

Exercise 3.3.5.19

Exercise 3.3.5.20

3.4 Null space and image

Exercise 3.4.3.1

Exercise 3.4.3.2

Exercise 3.4.3.3

Exercise 3.4.3.4

Exercise 3.4.3.5

Exercise 3.4.3.6

Exercise 3.4.3.7

Exercise 3.4.3.8

Exercise 3.4.3.9

Exercise 3.4.3.10

Exercise 3.4.3.11

Exercise 3.4.3.12

Exercise 3.4.3.13

Exercise 3.4.3.14

Exercise 3.4.3.15

Exercise 3.4.3.16

Exercise 3.4.3.17

Exercise 3.4.3.18

Exercise 3.4.3.19

Exercise 3.4.3.20

Exercise 3.4.3.21

Exercise 3.4.3.22

Exercise 3.4.3.23

Exercise 3.4.3.24

Exercise 3.4.3.25

3.5 Span and linear independence

Exercise 3.5.4.1

Exercise 3.5.4.2

Exercise 3.5.4.3

Exercise 3.5.4.4

Exercise 3.5.4.5

Exercise 3.5.4.6

Exercise 3.5.4.7

Exercise 3.5.4.8

Exercise 3.5.4.9

Exercise 3.5.4.10

Exercise 3.5.4.11

Exercise 3.5.4.12

Exercise 3.5.4.13

Exercise 3.5.4.14

Exercise 3.5.4.15

Exercise 3.5.4.16

Exercise 3.5.4.17

Exercise 3.5.4.18

Exercise 3.5.4.19

Exercise 3.5.4.20 Span, independence, and invertibility

Exercise 3.5.4.21 Linear transformations, span, and independence

3.6 Bases

Exercise 3.6.3.1

Exercise 3.6.3.2

Exercise 3.6.3.3

Exercise 3.6.3.4

Exercise 3.6.3.5

Exercise 3.6.3.6

Exercise 3.6.3.7

Exercise 3.6.3.8

Exercise 3.6.3.9

Exercise 3.6.3.10

Exercise 3.6.3.11

Exercise 3.6.3.12

Exercise 3.6.3.13

Exercise 3.6.3.14 Bases for important matrix subspaces

Exercise 3.6.3.15

Exercise 3.6.3.16

Exercise 3.6.3.17

Exercise 3.6.3.18

Exercise 3.6.3.19

Exercise 3.6.3.20

Exercise 3.6.3.21

Exercise 3.6.3.22

3.7 Dimension

Exercise 3.7.2.1

Exercise 3.7.2.2

Exercise 3.7.2.3

Exercise 3.7.2.4

Exercise 3.7.2.5

Exercise 3.7.2.6

Exercise 3.7.2.7

Exercise 3.7.2.8

Exercise 3.7.2.9

Exercise 3.7.2.10

Exercise 3.7.2.11

Exercise 3.7.2.12

Exercise 3.7.2.13

Exercise 3.7.2.14

Exercise 3.7.2.15

Exercise 3.7.2.16 Dimensions of important matrix subspaces

Exercise 3.7.2.17

3.8 Rank-nullity theorem and fundamental spaces

Exercise 3.8.4.1

Exercise 3.8.4.2

Exercise 3.8.4.3

Exercise 3.8.4.4

Exercise 3.8.4.5

Exercise 3.8.4.6

Exercise 3.8.4.7

Exercise 3.8.4.8

Exercise 3.8.4.9

Exercise 3.8.4.10

Exercise 3.8.4.11

Exercise 3.8.4.12

Exercise 3.8.4.13

Exercise 3.8.4.14

Exercise 3.8.4.15

Exercise 3.8.4.16

Exercise 3.8.4.17

Exercise 3.8.4.18

Exercise 3.8.4.19

Exercise 3.8.4.20

Exercise 3.8.4.21

4.1 Inner product spaces

Exercise 4.1.5.1

Exercise 4.1.5.2

Exercise 4.1.5.3

Exercise 4.1.5.4

Exercise 4.1.5.5

Exercise 4.1.5.6

Exercise 4.1.5.7

Exercise 4.1.5.8

Exercise 4.1.5.9

Exercise 4.1.5.10

Exercise 4.1.5.11

Exercise 4.1.5.12

Exercise 4.1.5.13

Exercise 4.1.5.14

Exercise 4.1.5.15

Exercise 4.1.5.16

Exercise 4.1.5.17

Exercise 4.1.5.18 Isometries of inner product spaces

4.2 Orthogonal bases

Exercise 4.2.3.1

Exercise 4.2.3.2

Exercise 4.2.3.3

Exercise 4.2.3.4

Exercise 4.2.3.5

Exercise 4.2.3.6

Exercise 4.2.3.7

Exercise 4.2.3.8

Exercise 4.2.3.9

Exercise 4.2.3.10

Exercise 4.2.3.11

Exercise 4.2.3.12

Exercise 4.2.3.13

Exercise 4.2.3.14

4.3 Orthogonal projection

Exercise 4.3.6.1

Exercise 4.3.6.2

Exercise 4.3.6.3

Exercise 4.3.6.4

Exercise 4.3.6.5

Exercise 4.3.6.6

Exercise 4.3.6.7

Exercise 4.3.6.8

Exercise 4.3.6.9

Exercise 4.3.6.10 Dimension of

W^{⊥}

Exercise 4.3.6.11

Exercise 4.3.6.12

Exercise 4.3.6.13

5.1 Coordinate vectors

Exercise 5.1.3.1

Exercise 5.1.3.2

Exercise 5.1.3.3

Exercise 5.1.3.4

Exercise 5.1.3.5

Exercise 5.1.3.6

Exercise 5.1.3.7

Exercise 5.1.3.8

Exercise 5.1.3.9

Exercise 5.1.3.10

Exercise 5.1.3.11

Exercise 5.1.3.12

Exercise 5.1.3.13 Orthonormal coordinate vectors

5.2 Matrix representations of linear transformations

Exercise 5.2.4.1

Exercise 5.2.4.2

Exercise 5.2.4.3

Exercise 5.2.4.4

Exercise 5.2.4.5

Exercise 5.2.4.6

Exercise 5.2.4.7

Exercise 5.2.4.8

Exercise 5.2.4.9

Exercise 5.2.4.10

5.3 Change of basis

Exercise 5.3.5.1

Exercise 5.3.5.2

Exercise 5.3.5.3

Exercise 5.3.5.4

Exercise 5.3.5.5

Exercise 5.3.5.6

Exercise 5.3.5.7

Exercise 5.3.5.8

Exercise 5.3.5.9

Exercise 5.3.5.10

Exercise 5.3.5.11

Exercise 5.3.5.12

Exercise 5.3.5.13

Exercise 5.3.5.14

Exercise 5.3.5.15 Determinant of orthogonal matrices

Exercise 5.3.5.16

Exercise 5.3.5.17

Exercise 5.3.5.18

Exercise 5.3.5.19 Reflection in

R^{2}

5.4 Eigenvectors and eigenvalues

Exercise 5.4.4.1

Exercise 5.4.4.2

Exercise 5.4.4.3

Exercise 5.4.4.4

Exercise 5.4.4.5

Exercise 5.4.4.6

Exercise 5.4.4.7

Exercise 5.4.4.8

Exercise 5.4.4.9

Exercise 5.4.4.10

Exercise 5.4.4.11

Exercise 5.4.4.12

Exercise 5.4.4.13

Exercise 5.4.4.14

5.5 Diagonalization

Exercise 5.5.5.1

Exercise 5.5.5.2

Exercise 5.5.5.3

Exercise 5.5.5.4

Exercise 5.5.5.5

Exercise 5.5.5.6

Exercise 5.5.5.7

Exercise 5.5.5.8

Exercise 5.5.5.9

Exercise 5.5.5.10

Exercise 5.5.5.11

Exercise 5.5.5.12

Exercise 5.5.5.13

Exercise 5.5.5.14

Exercise 5.5.5.15

Exercise 5.5.5.16

Exercise 5.5.5.17

Exercise 5.5.5.18

5.6 The spectral theorem

Exercise 5.6.3.1

Exercise 5.6.3.2

Exercise 5.6.3.3

Exercise 5.6.3.4

Exercise 5.6.3.5

Exercise 5.6.3.6

Exercise 5.6.3.7

Exercise 5.6.3.8

Exercise 5.6.3.9

Exercise 5.6.3.10

Exercise 5.6.3.11

Exercise 5.6.3.12

Exercise 5.6.3.13

Exercise 5.6.3.14

Exercise 5.6.3.15

Exercise 5.6.3.16

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