Bases of Subspaces

• Bases of a subspace is a set of vectors that can represent any vector in the subspace as a linear combination of itself.
• A basis for a subspace $\mathbb{S}$ of ${\mathbb{R}}^n$ is a linearly independent spanning set for $\mathbb{S}$.
• A subspace does not have a unique basis.
• The ${\vec{e}}_i$ vectors $\{{\vec{e}}_1,...,{\vec{e}}_n\}$ are called the standard basis for $R^n$

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If we have a subspace $\mathbb{S}$ of ${\mathbb{R}}^n$ and another set $B\subseteq \mathbb{S}$ And:

1. $B$ is linearly independent.
2. $\mathbb{S}=\text{Span }B$.

Then, $B$ is a basis for $\mathbb{S}$.

If $\mathbb{S}=0$, we define $B=\varphi$, the empty set to be the basis for $\mathbb{S}$.