Basis

In linear algebra, a basis of a subspace $V$ is a set of linearly independent vectors that span $V$. That was a bit much, so here’s a summary:

• $\text{span}({\vec{u}}_1,\dots ,{\vec{u}}_m)=V$.
• ${\vec{u}}_1,\dots ,{\vec{u}}_m$ are linearly independent.

The dimension of $V$ is the number of vectors in a basis of $V$.

Theorems

• Every spanning set is larger or equal to every linearly independent vector in $V$.
• All bases of a subspace $V\in {\mathbb{R}}^n$ consist of the same number of vectors.
• The vectors ${\vec{v}}_1,\dots ,{\vec{v}}_m$ in a subspace $V\in {\mathbb{R}}^n$ form a basis of $V$ if and only if every vector $\vec{v}\in V$ can be expressed uniquely as a linear combination $\vec{v}=c_1{\vec{v}}_1+\cdots +c_m{\vec{v}}_m$.