Basis

Definition

Let $V$ be a vector space and $\beta$ be a subset of $V$.
If $\beta$ is linearly independent and spans $V$, then $\beta$ is a basis of $V$.
We can also say that $\beta$ forms a basis for $V$.

Theorems

Theorem 1

Let $V$ be a vector space over a field $F$, and $\mathbf{u}\_1,\cdots ,\mathbf{u}\_n$ be $n$ distinct vectors of $V$.
Then $\beta =\{\mathbf{u}\_1,\cdots ,\mathbf{u}\_n\}$ forms a basis for $V$ if and only if an arbitrary vector $\mathbf{v}\in V$ can be uniquely written as a linear combination of vectors of $\beta$.
In other words, $\exists !a\_1,\cdots ,a\_n\in F$ such that $\mathbf{v}=\sum ^n\_i=1a\_i\mathbf{u}\_i$.

Proof