## 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:

associativity.
commutativity.
vector-vector distributivity.
scalar-scalar distributivity.
scalar-scalar multiplication compability.
neutral (identity) element of scalar multiplication.

### 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:

positive definiteness.
triangle relation.

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:

positive definiteness.
unique 0.
skew symmetry.
linearity in the first argument.

### Cauchy-Schwarz inequality

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:

...