A vector is anything that is closed under addition and scalar multiplication. In other words, it satisfies the following.

1. .
2. for some scalar .

Vector Space#

Vector space is a set of vectors and scalars that's defined under two operations, vector-vector addition and vector-scalar multiplication, such that the following properties hold for :

1. Closure: and .
2. Commutativity: .
3. Associativity: .
4. Distribution: , and .
5. Identity: there exists vector such that .
6. Additive inverse: for all , there exists a such that .

The dimension of this vector space is the maximal number of linearly independent (see below) vectors in .

Subspace#

Vector subspaces are sets that are themselves a vector space. Note that must lie in every subspace since it must be valid for .

The sum of subspaces and is defined as

Note that this is different from union, as there exists that's not in either subspace.

If we have and , then this is a direct sum decomposition, written as

Next, the orthogonal complement of is defined as

where denotes 📐 Angle > Orthogonality.

Linear Independence#

A linear combination of vectors is defined as

Vectors are linearly independent if there are no non-trivial () solutions to . If there is such a solution, then the vectors are linearly dependent.

Basis#

For a vector space and set of vectors , is a generating set if every vector can be expressed as a linear combination of . All linear combinations of is called the span of . Finally, if are linearly independent, is minimal and a basis of .

Every vector space has infinitely many bases, but they all have the same number of basis vectors. This number is the dimension of .

For a 🍱 Matrix , the number of linearly independent rows or columns is called the rank of . Note that the number of linearly independent rows equals the number of linearly independent columns.