## Vector

A vector is an ordered tuple of cells.

We write vectors as column vectors by default, that is:

## Vector space

A vector space is any set of vectors (vectors are in bold) together with two operators, + and ⋅ (note: λ and ω are any scalars), where the following conditions hold always:

### Subspace

S⊆V is a subspace iff .

(K being the set which contains the components of the vector)

### Linear independence

n vectors are *linearly dependent* iff there exist scalars so that:

AND

*not all of them*are 0 at the same time.

### Basis

Any n linear independent vectors form a basis of a vector space. n is the *cardinality* (dim) of the vector space and is constant for the vector space.

### Coordinate transformations

A (traditional) vector **v** can be transformed into coordinates (X,Y,Z) in the coordinate system denoted by the Basis by solving:

The components are called *contravariant components* of the vector **v**.

### Direct Sum

The vector space V is a direct sum of the subspaces S and T

... iff for every ∃ *unique* and so that .

### Norm

A norm is a function so that in a vector space V over K:

Every norm induces a function , called the *distance*.

### Inner product

A inner product is a function so that in a vector space V over K:

### Cauchy-Schwarz inequality

This leads to the angle φ between two vectors **u** and **v**:

### Orthogonality

Two vectors **u** and **v** are orthogonal iff:

Shorthand: u⊥v

### Orthonormality

Two vectors **u** and **v** are orthonormal iff they are orthogonal and:

#### Orthonormal Basis

A orthonormal basis is a basis where all vectors are orthonormal to each other.

#### Gram-Schmidt

The Gram-Schmidt algorithm can be used to complete a set of linearly independent vectors to a orthonormal basis.

Let be a set of linearly independent vectors.

Then one can calculate a set of vectors to form an orthogonal system where all the vectors are orthogonal to each other:

...