A basis of $V$ is a list of vectors in $V$ that is linearly independent and spans $V$. Every $v\in V$ must be _uniquely_ written as a linear combination of the list. Every list that spans $V$ can be converted into a basis of $V$ with a subset of the list—simply remove $v_{i}$ if $v_{i}\not\in \text{span}(v_{1}\ldots v_{i-1}, v_{i+1}, \ldots v_{n})$ and repeat until you can no longer do this. Similarly you can extend a linearly independent list into a basis of $V$ by continuously by adding $v_{i}s that are not in the span of the linearly independent list. > [!Insight] > If $V$ is a fininite dimensional subspace and $U$ is a subspace of $V$. Then there exists a subspace $W$ of $V$ such that $U\oplus W=V$. Additionally, we know that from [[1C3 - Problem Set|problem 1C.19]] that this $W$ is not necessarily unique.