Mathematics of Geometry, Mappings

Question 1

What is this set called?

Answer 1

It is the range of f.

Question 2

Give A and B in y = Ax + B if the equation represents a pure scaling.

Answer 2

B = 0. A is the matrix

q 0

0 p

Zero except on the diagonal.

Question 3

What is a figure?

Answer 3

A figure is a set of points in Euclidean space, which corresponds to a set of vectors.

We can say each point P is an element of the figure.

Question 4

Name and define this.

Answer 4

This is the two-dimensional vector space of real numbers.

It is the collection of all vectors describing all points in E2.

Question 5

What is this?

Answer 5

The Euclidean Plane.

Question 6

What kind of transformation is this?

Answer 6

This is a translation, a type of affine transformation.

Question 7

When is this mapping linear?

Answer 7

When the following equality holds.

Question 7

Name two non-affine transformations.

Answer 7

Reflection and shear.

Question 8

What is this?

Answer 8

This is the composition of f and g.

(g o f)(a) is g(f(a)) for any a in A.

If f maps A to B and g maps B to C then (g o f) maps A to C.

Question 9

If the determinant of matrix A is zero, is A invertible?

Answer 9

A is not invertible if its determinant is zero.

Conversely, if A’s determinant is nonzero, it is invertible.

Question 9

Is a projection transformation linear?

Answer 9

Yes, a projection transformation is linear but non-affine.

Question 10

What do we know about the composition of two bijections?

Answer 10

The composition of two bijections is also a bijection.

Question 13

If A is invertible, what kind of transformation is this?

Answer 13

This is an affine transformation.

An affine transformation of the plan has an inverse which is also an affine transformation.

Question 14

When is this surjective?

Answer 14

When f(A)=B

Which says that every element of B is the image of at least one element of A.

Also called onto

Question 14

What is this?

Answer 14

This mapping is the inverse of f if the following is true for every a in A.

Question 14

What do we know about the composition of affine transformations?

Answer 14

The composition of affine transformations is also an affine transformation.

Question 15

How is this defined if sigma is a figure?

Answer 15

Question 17

If x and y on the 2D coordinate plane are to be represented by homogeneous coordinates (a,b,w), name a translation that will represent infinity in 2D coordinates.

Answer 17

x=a/w

y=a/w

So the point (1,1,0) is one way to represent infinity

Homogeneous coordinates add a concept of infinity to Euclidean coordinates, which lack them.

Question 18

When is this injective?

Answer 18

When each element in the range of f is the image of exactly one element in A.

When a set is injective, f(x)=f(y) implies x=y.

Question 18

Define the rotation matrix in this rotation transformation.

Answer 18

Question 20

What are the terms for A and B in this?

Answer 20

A is the domain of f. B is the codomain of f.

Question 21

Name the properties of an affine transformation

Answer 21

(1) maps a line to a line,
(2) maps a line segment to a line segment,
(3) preserves the property of parallelism among lines and line segments, (4) maps an n-gon to an n-gon,
(5) maps a parallelogram to a parallelogram,
(6) preserves the ratio of lengths of two parallel segments, and
(7) preserves the ratio of areas of two figures.

Question 22

What is a singular matrix?

Answer 22

A matrix that is not invertible.

Question 23

What is the transformation of a set?

Answer 23

It is the bijection of a set unto itself.

Each transformation of the Euclidean Plane corresponds to a transformation of the 2D vector space of real numbers.

If f(P)=P’, then

f~() =

Both point and vector transformations are allowed.

Question 24

How is a vector space generally defined?

Answer 24

A vector space is ***any*** collection of objects on which addition and multiplication are defined such that they have the following properties: Addition must be a closure, associative and commutative laws are obeyed, and additive inverse and identity exist. Multiplication must meet the following requirements with scalars: must be a closure, unitary, associative and both kinds of distributive laws must be obeyed

Question 26

State the fundamental theorem of affine transformations

Answer 26

Given two ordered sets of three noncollinear points each, there exists an affine mapping f between those two sets. And affine transformation between sets of points preserves collinearity and distance ratios.

Question 27

Give A and B in y = Ax + B if the equation represents a pure rotation.

Answer 27

B = 0. A is the matrix cos(θ) sin(-θ) sin(θ) cos(θ)

Question 28

What does OXY denote?

Answer 28

A coordinate system (not necessarily cartesian) with the following axes.

Question 29

Give A and B in y = Ax + B if the equation represents a pure translation.

Answer 29

A is I, and B is the translation coordinates.

Question 30

What is meant by this?

Answer 30

This is the **function** or **mapping** from A to B. It consists of the set of ordered pairs (a,b) where a is an element of A and b is an element of B, with the following properties: For every a, there exists a unique b such that (a,b) is an element of the mapping f. Another denotation: f(a)=b. Still another denotation:

Question 31

When is T a linear mapping?

Answer 31

When T(**u**+**v**) = T(**u**) + T(**v**) and T(c**u**) = cT(**u**)

Question 32

What are the terms for a and b in this?

Answer 32

b is the **image** of a in is f. a is the **preimage** of b in f.

Question 33

When is this **bijective**?

Answer 33

When it is both *injective* and *surjective*.