# Basis **A [[subset]] of a [[vector space]] which can be linearly combined to form every element of the vector space.** ![[Basis.svg]] *A basis can represent every vector of its vector space (left); conversely, every vector can be represented by multiple bases (right)* A **basis** is a [[subset]] of the elements of a [[vector space]] $V$ which can be used in a *finite linear combination* to uniquely express every element of $V$. The coefficients of the linear combination are known as the *components* or *coordinates* and the elements are known as the *basis vectors*. > [!NOTE] Basis > A subset of the elements of a vector field $V$ over a field $F$ is a **basis** $B$ if they are: > - *[[Linear Independence|linearly independent]]* - for every finite subset $\{\mathbf{v}_{1},\dots,\mathbf{v}_{n}\}$ of $B$, if $\lambda_{1}\mathbf{v}_{1}+\cdots+\lambda_{n}\mathbf{v}_{n}=0$ only if $\lambda_{1}=\cdots=\lambda_{n}=0$ > - a *[[Span|spanning set]]* - for every element $\mathbf{v}$ in $V$, $\mathbf{v}=\lambda_{1}\mathbf{v}_{1}+\cdots+\lambda_{n}\mathbf{v}_{n}$ for some $\lambda_{1},\dots,\lambda_{n}\in F$ and $\mathbf{v}_{1},\dots,\mathbf{v}_{n}\in B$ A vector space may have several bases, although they all have the same number of elements which is referred to as the *dimension* of the vector space. ## Standard basis The **standard basis**, also known as the **natural** or **canonical basis**, is the set of vectors whose components are *all zero except for one* which is equal to $1$. $\mathbf{e}_{1}=\left(\begin{matrix}1\\ 0\\0\\ \vdots\end{matrix}\right)\quad\mathbf{e}_{2}=\left(\begin{matrix}0\\ 1\\0\\ \vdots\end{matrix}\right)\quad\mathbf{e}_{3}=\left(\begin{matrix}0\\ 0\\1\\ \vdots\end{matrix}\right)\quad\dots$ <br> ![[VectorScalarComponentsStandardBasis.svg|350]] *A set of standard basis vectors of a vector space with an arbitrary number of dimensions (top); a vector $\mathbf{a}$ represented using the standard basis vectors (bottom)*