🏹 Vector

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

1. .
2. for some scalar .

Info

Geometric vectors, polynomials, audio signals, and tuple elements of are all vectors.

Vector Spaces §

Vector space is a set that’s closed under addition and scalar multiplication. It also contains the zero vector . Examples of vector spaces include and .

Vector subspaces are sets that are closed under addition and scalar multiplication.

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.

Bases §

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.