9) Euclidean Space and Isometries Flashcards
How should vectors be handled with respect to a basis in Rn
How does changing the basis affect the matrix representation of a linear transformation T in Rn
What is the inner product of two vectors
What is the modulus of a vector
When are a set of vectors orthogonal and orthonormal
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
What is the distance between two vectors
What are the properties of distance
What does it mean for an isomoprhism to be orientation preserving
What is an isometry
What conditions are equivalent for a linear transformation to be an isometry
What is a special orthogonal matrix
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)
What is a translation
What is an affine transformation
A transformation of the form x → Ax + b
Are isometries affine transformations
Yes, isometries are affine
What are the classification of 2D isometries
Give an example of a glide reflection
When is an affine transformation orientation preserving
What is the Gram-Schmidt process
How does the orthogonal group O(n) act on the (n−1)-dimensional unit sphere S^n−1
What is the unit sphere
The set of unit vectors in R^n is called the unit sphere and is typically denoted by S^n−1
What property do all subspaces of R^n have
What is the image of f restriced to U
What does it mean if U is f-invariant
What is the subspace perpendicular to U
What are the properties of a subspace U and the subspace perpendicular to U
What happens to the subspace perpendicular to U under a linear isometry if U is invariant
Describe the proof that if U is T-invariant then so is U^⊥
Describe the classification of orthogonal 2 × 2 matrices
Describe the classification of orthogonal 3 × 3 matrices
What are the eigenvalue properties of matrices in O(2) and O(3) based on their determinants
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
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
What is a reflection
Describe the graph that represents a reflection
What is a hyperplane in R^N perpendicular to N through cN
What is a reflection in Π (hyperplane)
Describe the graph for a reflection in Π (hyperplane)
Describe the graph of a glide reflection in Π (hyperplane)
What is a glide reflection in a Π (hyperplane)
Describe the classification of isometries of R^2
What is the type of an isometry in R^2
The type of an isometry in R^2 is either -
* Identity
* Translation
* Rotation
* Reflection
* Glide reflection
Can an isometry have more than one type
No, f can only be one type not two different types simultaneously
Describe the classification of isometries of R^3
What property do conjugate isometries have
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
What is the fixed point set of g
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
Describe the proof that g. fix(f) = fix(gfg^−1) and if S is f-invariant then g.S is gfg^−1 invariant
Under what conditions are two isometries f and g in E(2) considered conjugate
How do the determinant, trace, and fixed points of different isometry types in E(2) compare
