Vector space

The only way to study a vector space is to find a basis for it.

Basis

A collection of vectors $v_1,\dots ,v_k\in V$ is a basis for $V$ if any vectors in $V$ is a UNIQUE linear combination of them. A vector space has $k$ dimension if it has a basis made of $k$ vectors.

Given a vector space $V$ and a basis $B=\{v_1,\dots ,v_n\}$, then for any vector $v\in V$, if $v=a_1{v}_1+\cdots +a_n{v}_n$, then we say the coordinates of $v$ under basis $B$ is $v_B=\left(\begin{array}{c}{a}_1\\ \dots \\ a_n\end{array}\right)$.