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Groups and Geometry > 9) Euclidean Space and Isometries > Flashcards

9) Euclidean Space and Isometries Flashcards

1
Q

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

A
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2
Q

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

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3
Q

What is the inner product of two vectors

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4
Q

What is the modulus of a vector

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5
Q

When are a set of vectors orthogonal and orthonormal

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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

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6
Q

What is the distance between two vectors

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7
Q

What are the properties of distance

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8
Q

What does it mean for an isomoprhism to be orientation preserving

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9
Q

What is an isometry

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10
Q

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

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11
Q

What is a special orthogonal matrix

A

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)

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12
Q

What is a translation

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13
Q

What is an affine transformation

A

A transformation of the form x → Ax + b

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14
Q

Are isometries affine transformations

A

Yes, isometries are affine

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15
Q

What are the classification of 2D isometries

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16
Q

Give an example of a glide reflection

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17
Q

When is an affine transformation orientation preserving

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18
Q

What is the Gram-Schmidt process

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19
Q

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

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20
Q

What is the unit sphere

A

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

21
Q

What property do all subspaces of R^n have

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22
Q

What is the image of f restriced to U

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23
Q

What does it mean if U is f-invariant

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24
Q

What is the subspace perpendicular to U

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25
Q

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

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26
Q

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

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27
Q

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

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28
Q

Describe the classification of orthogonal 2 × 2 matrices

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29
Q

Describe the classification of orthogonal 3 × 3 matrices

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30
Q

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

A

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

31
Q

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

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32
Q

What is a reflection

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33
Q

Describe the graph that represents a reflection

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34
Q

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

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35
Q

What is a reflection in Π (hyperplane)

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36
Q

Describe the graph for a reflection in Π (hyperplane)

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37
Q

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

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38
Q

What is a glide reflection in a Π (hyperplane)

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39
Q

Describe the classification of isometries of R^2

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40
Q

What is the type of an isometry in R^2

A

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

41
Q

Can an isometry have more than one type

A

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

42
Q

Describe the classification of isometries of R^3

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43
Q

What property do conjugate isometries have

A

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

44
Q

What is the fixed point set of g

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45
Q

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

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46
Q

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

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47
Q

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

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48
Q

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

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Groups and Geometry (12 decks)

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