9) Euclidean Space and Isometries

Question 1

How should vectors be handled with respect to a basis in Rn

Answer 1

Question 2

How does changing the basis affect the matrix representation of a linear transformation T in Rn

Answer 2

Question 3

What is the inner product of two vectors

Answer 3

Question 4

What is the modulus of a vector

Answer 4

Question 5

When are a set of vectors orthogonal and orthonormal

Answer 5

Orthogonal - if whenever we have u, v ∈ S with u ≠ v then u · v = 0. Furthermore, S is orthonormal if in addition we have for all u ∈ S we have |u| = 1

Question 6

What is the distance between two vectors

Answer 6

Question 7

What are the properties of distance

Answer 7

Question 8

What does it mean for an isomoprhism to be orientation preserving

Answer 8

Question 9

What is an isometry

Answer 9

Question 10

What conditions are equivalent for a linear transformation to be an isometry

Answer 10

Question 11

What is a special orthogonal matrix

Answer 11

A matrix A is special orthogonal if it is an orthogonal matrix and
det A = 1
The set of special orthogonal n × n matrices is denoted by SO(n)

Question 12

What is a translation

Answer 12

Question 13

What is an affine transformation

Answer 13

A transformation of the form x → Ax + b

Question 14

Are isometries affine transformations

Answer 14

Yes, isometries are affine

Question 15

What are the classification of 2D isometries

Answer 15

Question 16

Give an example of a glide reflection

Answer 16

Question 17

When is an affine transformation orientation preserving

Answer 17

Question 18

What is the Gram-Schmidt process

Answer 18

Question 19

How does the orthogonal group O(n) act on the (n−1)-dimensional unit sphere S^n−1

Answer 19

Question 20

What is the unit sphere

Answer 20

The set of unit vectors in R^n is called the unit sphere and is typically denoted by S^n−1

Question 21

What property do all subspaces of R^n have

Answer 21

Question 22

What is the image of f restriced to U

Answer 22

Question 23

What does it mean if U is f-invariant

Answer 23

Question 24

What is the subspace perpendicular to U

Answer 24

Question 25

What are the properties of a subspace U and the subspace perpendicular to U

Answer 25

Question 26

What happens to the subspace perpendicular to U under a linear isometry if U is invariant

Answer 26

Question 27

Describe the proof that if U is T-invariant then so is U^⊥

Answer 27

Question 28

Describe the classification of orthogonal 2 × 2 matrices

Answer 28

Question 29

Describe the classification of orthogonal 3 × 3 matrices

Answer 29

Question 30

What are the eigenvalue properties of matrices in O(2) and O(3) based on their determinants

Answer 30

We have the following 1. If A ∈ O(2) and det A = −1, then A has −1 as an eigenvalue 2. If A ∈ O(3) and det A = 1, then A has 1 as an eigenvalue 3. If A ∈ O(3) and det A = −1, then A has −1 as an eigenvalue

Question 31

How can an orthogonal matrix A in O(n) be represented in a specific orthonormal basis if one eigenvector N is such that |N| = 1

Answer 31

Question 32

What is a reflection

Answer 32

Question 33

Describe the graph that represents a reflection

Answer 33

Question 34

What is a hyperplane in R^N perpendicular to N through cN

Answer 34

Question 35

What is a reflection in Π (hyperplane)

Answer 35

Question 36

Describe the graph for a reflection in Π (hyperplane)

Answer 36

Question 37

Describe the graph of a glide reflection in Π (hyperplane)

Answer 37

Question 38

What is a glide reflection in a Π (hyperplane)

Answer 38

Question 39

Describe the classification of isometries of R^2

Answer 39

Question 40

What is the type of an isometry in R^2

Answer 40

The type of an isometry in R^2 is either - * Identity * Translation * Rotation * Reflection * Glide reflection

Question 41

Can an isometry have more than one type

Answer 41

No, f can only be one type not two different types simultaneously

Question 42

Describe the classification of isometries of R^3

Answer 42

Question 43

What property do conjugate isometries have

Answer 43

Let f, g ∈ E(n), where n = 2 or n = 3. If f and g are conjugate, then f and g have the same isometry type

Question 44

What is the fixed point set of g

Answer 44

Question 45

How does the action of a group element g in G affect the fixed points and invariant sets of another group element f in G

Answer 45

Question 46

Describe the proof that g. fix(f) = fix(gfg^−1) and if S is f-invariant then g.S is gfg^−1 invariant

Answer 46

Question 47

Under what conditions are two isometries f and g in E(2) considered conjugate

Answer 47

Question 48

How do the determinant, trace, and fixed points of different isometry types in E(2) compare

Answer 48