Introduction to Vector Space

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# 👋 Introduction to Vector Space

Wai-Shing Luk

2018-09-12 📅

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### Vector Space

A vector space over a field $F$ is a set $V$ together with 2 binary operators that satisfy 8 axioms below:

- Associativity of addition
    $$u + (v + w) = (u + v) + w$$
- Commutativity of addition
    $$v + w = w + v$$
- Identity element of addition
    - There exists an element $0$ such that $v + 0 = v$ for all $v \in V$
- Inversion elements of addition
    - There exist an element $w$, denoted by $-v$, such that $v + w = 0$

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### Vector Space (cont'd)

- Distributivity of scalar multiplication w. r. t. vector addition
  $$\alpha \cdot (v + w) = \alpha \cdot v + \alpha \cdot w$$
- Distributivity of scalar multiplication w. r. t. field addition
 $$(\alpha + \beta)\cdot v = \alpha \cdot v + \beta \cdot v$$
- Compatibility of scalar multiplication w. r. t. field multiplication
  $$\alpha\cdot(\beta\cdot v) = (\alpha\cdot\beta)\cdot v$$
- Identity element of scalar multiplication
  $$1\cdot v = v,$$
  where $1$ denotes that multiplicative identity in $F$

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### Vector Space in C++ Concepts

```cpp
template <class V, typename F = typename V::value_type>
concept bool Vector_Space()
{
  return Equality_comparable<V>() &&
    requires (V v, V w, F a) {
    { V() } → V;   // zero element
    { -v } → V;    // negation
    { v + w } → V; // addition
    { a * v } → V; // multiplication
  };
}
```

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### Vector Space in C++ Concepts (Cont'd)

```cpp
  template <class V, class F = typename V::value_type>
    requires Vector_Space<V,F>()
  void Vector_Space_test(V v, V w, V u, F a, F b)
  {
    assert( v + w == w + v || !"commutativity of addition" );
    assert( u + (v + w) == (u + v) + w );
    assert( v + V(0) == v )
    assert( v + (-v) == V(0) );
    assert( a * (v + w) == a * v + a * w );
    assert( (a + b) * v == a * v + b * v );
    assert( a * (b * v) == (a * b) * v );
    assert( F(1) * v == v );
  }
```

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### Normed Space

$S$ is a normed space if there exists a norm function $\| \cdot \|$ that satisfies the following 3 axioms:

- $\| v \| \ge 0$ and $\| v \| = 0$ iff $v=0$
- $\| \alpha \cdot v \|$ = $|\alpha| \cdot \|v\|$ (uniformality)
- $\| v \| + \| w \| \ge \| v + w \|$

👉 Note:

1. $\|\cdot\|$ is always a convex function
2. A distance $d$ between two vectors $v$ and $w$ can be defined as $d(v, w) = \| v - w \|$.

Norm examples: Euclidean norm ($L_2$-norm), Manhattan norm ($L_1$-norm)

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### Normed space in C++ Concept

```cpp
  template <class M, class F = typename M::value_type>
  concept bool Normed_Space() {
    return Vector_Space<M,F>()
      && requires (M v) { { norm(v) } → F; };
  }

template <class M, class F = typename M::value_type>
        requires Normed_Space<M,F>()
  void Normed_Space_test(M v, M w, F a) {
    assert( norm(v) >= F(0) );
    assert( norm(M(0)) == F(0) );
    assert( norm(v) != F(0) || v == M(0) );
    assert( norm(a*v) == abs(a) * norm(v) || !"uniformality");
    assert( norm(v) + norm(w) >= norm(v + w) );
  }
```

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### Inner Product Space

$S$ is an inner product space if there exists an inner product function $\langle a, b\rangle$ that satisfies the following 3 axioms:

- Conjugate symmetry:
    $$\langle x, y\rangle = (\langle y,x\rangle)^*$$
- Linearity in the first argument:
    $$\langle\alpha \cdot x, y\rangle = \alpha \cdot \langle x, y\rangle$$
    $$\langle x + y, z\rangle = \langle x, z\rangle + \langle y, z\rangle$$

👉 Note:

1. we say $x$ perpendicular (or orthogonal) to $y$ if $\langle x, y\rangle = 0$
2. A norm function of a real vector can be defined as $\|x\| = \sqrt{\langle x, x\rangle}$

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### Inner Product Space in C++ Concept

```cpp
  template <class M, class F = typename M::value_type>
  concept bool Inner_Product_Space() {
    return Vector_Space<M,F>()
      && requires (M x, M y) { { dot(x, y) } → F; };
  }

template <class M, class F = typename M::value_type>
    requires Inner_Product_Space<M,F>()
  void Inner_Product_Space_test(M x, M y, M z, F a) {
    assert( dot(x, y) >= F(0) );
    assert( dot(M(0),M(0)) == F(0) );
    assert( dot(x,x) != F(0) || x == M(0) );
    assert( dot(x, y) == dot(y, x) );
    assert( dot(a*x, y) == a*dot(x,y) );
    assert( dot(x+y, z) == dot(x,z) + dot(y,z) );
  }
```

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### Cross Product

The cross product of 2 vectors $v$ and $w$, denoted by $v \times w$, is defined as a vector $u$ that is perpendicular to both $v$ and $w$, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

👉 Note:

-  $v \times w = -w \times v$ (anti-commutative)
-  $v \times w = 0$ if $v$ is parallel to $w$

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# Q&A 🎤