Representation in a Basis

Let $\mathcal{B} = \{ \vec{b}_1, \dots,\vec{b}_n \}$ be a basis for a subspace $V$ and let $\vec{v} \in V$.

The representation of $\vec{v}$ in the $\mathcal{B}$ basis, $[\vec{v}]_{\mathcal{B}}$, is the column matrix

$[\vec{v}]_{\mathcal{B}}=\begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix}$ where $a_1,\dots,a_n$ uniquely satisfy $\vec{v} = a_1\vec{b}_1+\dots+a_n\vec{b}_n$

$\begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix}_{\mathcal{B}}=a_1\vec{b}_1+\dots+a_n\vec{b}_n$

Notations

If a problem has only one basis, we can write $\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right]$ to mean $\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right]_{\mathcal{E}}$

If there are multiple bases, we will always write $\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right]_{\mathcal{X}}$

True Vectors vs Representations

• $\left[\text{true vector}\right]_{\mathcal{X}}=\text{list of numbers}$
• $[\text{list of numbers}]_{\mathcal{X}}=\text{true vector}$

Examples:

• $\vec{x}=2\vec{e}_1+3\vec{e}_2∈ℝ^2$ is a vector.
• $[\vec{x}]_{\mathcal{E}}=\left[\begin{smallmatrix} 2 \\ 3 \end{smallmatrix}\right]$ is a list of numbers.
• $\left[\begin{smallmatrix} 2 \\ 3 \end{smallmatrix}\right]_{\mathcal{E}}$ is a vector.
• $\vec{x}=\left[\begin{smallmatrix} 2 \\ 3 \end{smallmatrix}\right]{\mathcal{E}}≠[\vec{x}]{\mathcal{E}}$

Basis Orientation

For a ordered basis $\mathcal{B}=\{\vec{b}_1,\dots,\vec{b}_n\}$:

• Right-handed or Positively Oriented if it can be continuously transformed to $\mathcal{E}$ while being linearly independent the whole time
• Left-handed or Negatively Oriented if otherwise