:PROPERTIES: :ID: 6eec1bb7-1ca7-4073-a27f-5b76917668fa :mtime: 20231019010114 20231014015046 :ctime: 20231014014921 :END: #+title: vector basis #+filetags: :public:project: * Definition Given a list of [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][vectors]] $B = \{\vec{e}_{1},\vec{e}_{2}, \ldots , \vec{e}_{n} \}$ We say that $B$ is a *basis* of the vector space if and only if 1) $B$ is [[id:e65e7624-7323-40e2-8b8d-5fe498877e85][linearly independent]] 2) The number of vectors in $B$ is equal to the dimension of the vector space. * Properties of a basis $\vec{0}$ can never be a part of a basis. * Special Types of Bases If all the basis vectors are orthogonal to each other, we call that basis an [[id:c20fbea5-f4f3-483a-b695-0baabcaec598][orthogonal basis]] If all the basis vectors are orthogonal to each other, and all the basis vectors have [[id:1f92fe16-9870-4cb4-9c50-4f28fc02a805][Vector Magnitude]] 1, then we call taht basis an [[id:31ec01b7-1665-44e5-ba44-4957097fc880][Orthonormal]].