1Metric spaces

  • A space (a mathematical space) is a set of points.
  • A point is an element in a set.
  • "Metric" is another term for "distance".
    • We usually write \(d(x,y)\) to mean "the distance between point \(x\) and point \(y\)".
  • A metric space is a space and a metric.

  • Relationship:

    geometry set theory
    space set
    point element

2Normed vector spaces

  • An Euclidean vector is something with magnitude and direction.
  • A vector is an element of a vector space.
  • Understand scalar–vector multiplication:
    • If \(k\) is a scalar (a real number) and \(x\) is a vector, then \(kx\) is a vector whose direction is the same as \(x\) but whose length is \(k\) times the length of \(x\).
  • A vector space is a set of vectors plus some structures?
  • Notation
    • \(\norm{x}\) ("the norm of \(x\)") means the length of vector \(x\).
    • "Norm" means "length".
    • The norm of a vector is the length (the magnitude) of that vector.
  • Understand vector addition.
  • Understand vector negation.

  • Understand point–vector addition.

    operation left argument right argument result
    addition vector vector vector
    addition point vector point
    subtraction vector vector vector
    subtraction point point vector
  • Derive subtraction from addition and negation.
    • Subtraction can be derived from addition and negation: \(x-y = x+(-y)\).
    • Negation: \(-x\) is something such that \(x+(-x) = 0\).
    • If \(p+v=r\), then, subtract both sides by \(v\) to get \(p=r-v\).
  • See also a brief history of vectors.
  • An Euclidean vector space is a set of vectors with Euclidean metric. A metric is a function that maps each vector to its length. An Euclidean metric is distance as we know it in everyday situation.
  • Euclidean distance can be derived from the Pythagorean theorem that relates the lengths of the sides of a right triangle.

3Relationship between metric and norm

  • Because a vector can be obtained by subtracting two points:
    • A metric space can be derived from a normed vector space.

\[ d(x,y) = \norm{y-x} \]

4Relationship between spaces: embedding, projection, homeomorphism

  • Understand ambient spaces.
  • Understand embedding.
    • Example: A line may be embedded in a three-dimensional space.
  • Understand projection.
    • Example: A cube may be projected onto a plane.
    • Understand that shadows can be modeled by geometric projections.
  • Understand homeomorphism.

5Circles, pi, angles, radian

  • A circle with center \(c\) and radius \(r\) is \( \SetBuilder{x}{d(c,x) = r} \) (the set of every point whose distance from \(c\) is \(r\)).
  • \( \pi \) is the ratio of a circle's circumference to its diameter.
    • All circles have that ratio.
    • The first few digits of π are 3.141659…
    • π is sometimes approximated as 22/7.
  • An angle is …?
  • An angle may be thought of as one revolution, half revolution, quarter revolution, 3/4 revolution, etc., but radian is the natural unit for angles.
    • 360 degrees = 2π radian = 1 revolution
    • Understand why1 radian is the natural unit for angles.
  • Understand that a positive angle usually means counterclockwise rotation.
  • Circle, sector, angle, congruence
    • Consider a circle of radius \(r\).
    • Consider a sector of angle \(\alpha\) in that circle.
    • The circumference of that sector is \( r \cdot \alpha \).
    • An angle is the "radius-independent size" of a sector.

6Basic shapes

  • A polygon (an \(n\)-gon) is a shape with \(n\) vertices / edges / internal angles.
    • A vertex is a point.
    • An edge is a straight line that connects two vertices.
    • An internal angle is …
  • A triangle is a shape with three vertices / three edges / three internal angles.
    • triangle = three-angle
    • quadrangle = four-angle, etc.
  • A circle is \( \SetBuilder{x}{d(c,x) = r} \) (see above).

7Basic solids

  • A sphere is \( \SetBuilder{x}{d(c,x) = r} \).
    • A sphere is the surface of a ball.

8Right triangles and trigonometry

  • A right triangle is a triangle with one right angle.
    • The hypothenuse is the side across the right angle.
  • The Pythagorean theorem relates the length of the sides of every right triangle.
  • Trigonometry

9More dimensions

9.1Modeling physical space as a three-dimensional Euclidean space

There are two different meanings of "space". Physical space is often thought of as the volume occupied by objects.

Most likely, you already understand a three-dimensional Euclidean space, because it is your internal mental model of the space around you.

An Euclidean space is a set of points, not a set of coordinates.

We usually draw a point as a dot, but a point is a zero-dimensional geometric object that occupies no space at all.

This picture may help you visualize a three-dimensional real tuple space.

9.2Dimension

  • The number of dimensions of a space is the maximum number of mutually orthogonal lines that can coexist in that space.
    • For example, we know no way to arrange more than three mutually orthogonal rulers in physical space. Therefore, we think our physical space has three dimensions.
  • The number of dimensions of every subset of \( \Real^n \) is \(n\).
    • Do not confuse embeddings and subsets.
      • \( \Real^m \) is not a subset of \( \Real^n \) if \( m \neq n \).
    • We do not define \(\Real^0\).
    • We do not define the number of dimensions of the empty set.

How do we know how many dimensions a space has?

We know that a space has dimension \(n\) if we cannot simultaneously place more than \(n\) mutually orthogonal lines in that space. We infer that our space has three dimensions because we have not found how to arrange more than three mutually orthogonal lines in our space. For example, if we pick three rulers, we can arrange them so that they are orthogonal to each other. But, if we pick four rulers, we know no way to arrange them so that they are orthogonal to each other. That is why we think that space is three-dimensional.

9.3Orthogonality, perpendicularity

How do we know that two lines are orthogonal? By a protractor?

We know that two lines are orthogonal if all the four angles formed by their intersection are equal.

Other terms for "orthogonal" are "perpendicular" and "at a right angle".

9.4Hypersolids

  • polygon (many-corner) → polyhedron (many-seat) → polytope (many-place)
  • shape → solid → hypersolid
  • A line is \( \SetBuilder{x}{x = a + tb, ~ t \in \Real} \).
  • A hyperplane is \( \SetBuilder{x}{a \cdot x = b} \).
  • Understand two-dimensional geometric objects: line, circle, square.
  • Understand three-dimensional geometric objects: plane, sphere, cube.
  • Understand higher-dimensional geometric objects: hyperplane, hypersphere, hypercube, "hyper-" + <the name of a three-dimensional object>.

10Transformations, symmetries, and invariants

10.1Invariants

Let \(\phi : D \to C\) be a function.

Let \(T : D \to D\) be a function, usually called a "transformation".

We say "\(\phi\) is unaffected by \(T\)" or "\( \phi \) is \(T\)-invariant" or "\( T \) is an invariant (a symmetry) of \( \phi \)" iff, for all \(x \in D\): \[ \phi(x) = \phi(T(x)) \]

Here are some examples of invariants.

Let \( Tx = x + c \) represent translation.

Example: If \(\phi(x,o,r)\) means "\(x\) is a point on a circle with center \(o\) and radius \(r\)", then \(\phi(x,o,r) = \phi(Tx,To,r)\). (A predicate is a function whose codomain is the set of booleans.)

Example: If \(V\) is a vector space, then \(\SetBuilder{Tv}{v \in V} = V\). If we follow the "auto-lifting" convention, we can write the equation more prettily as \( TV = V \).

Example: Even functions exhibit mirror symmetry. (A function \( f \) is even iff \( f(x) = f(-x) \) for all applicable \(x\).)

Example: Periodicity is a special case of translation-invariance. (A function \(f\) has period \(p\) iff \(p\) is the smallest positive number such that \( f(x+p) = f(x) \) for all applicable \(x\). A function is periodic iff it has a period.)

10.2Symmetries

Example: let \(S\) be the set of points of an unlabeled square.

Let \(T\) be a rotation about the square's center by a right angle.

We write \(TS\) to mean "the result of rotating \(S\) about its center by a right angle".

We write \(TS = S\) to mean "rotating \(S\) about its center by a right angle produces \(S\) itself".

  1. <2020-01-26>↩