Basis

A basis for a vector space is a set of vectors that:

1. Spans the space (every vector can be written as a combination)
2. Is linearly independent (no redundancy)

Standard Basis

In $\mathbb{R}^2$: $\left\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\right\}$

In $\mathbb{R}^3$: $\left\{\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}\right\}$

Key Properties

• Every basis has the same number of vectors = dimension
• $\mathbb{R}^n$ has dimension $n$
• Any $n$ independent vectors in $\mathbb{R}^n$ form a basis
• The basis is not unique — infinitely many bases exist