---
Created: 2025-08-22
Type: Zettel
aliases:
References:
Links:
- "[[Vectors]]"
- "[[Basis]]"
tags:
- MATH31AH
---
- In $\mathbb{R}^n$, there are two standard [[basis]] [[Vectors]], $\vec{e}_{1}$ and $\vec{e}_{2}$
- in $\mathbb{R}^3$, there are three:
$$
\text{in } \mathbb{R}^n : \vec{e}_{1}=\begin{bmatrix}
1 \\ o
\end{bmatrix},
\vec{e}_{2} = \begin{bmatrix}
0 \\ 1
\end{bmatrix};
\text{ in } \mathbb{R}^3 : \vec{e}_{1} = \begin{bmatrix}
1 \\ 0 \\ 0
\end{bmatrix},
\vec{e}_{2} =
\begin{bmatrix}
0 \\ 1 \\ 0
\end{bmatrix},
\vec{e}_{3} =
\begin{bmatrix}
0 \\ 0 \\ 1
\end{bmatrix}.
$$
- This pattern of vectors continues all that way up to $\mathbb{R}^n$
- **Definition: Standard basis vectors.** The standard basis vectors in $\mathbb{R}^n$ are the vectors $\vec{e}_{j}$ with $n$ entries, the $j$th entry 1 and the others zero.
- The notation $\vec{i}, \vec{j}, \vec{k}$ is also used elsewhere
- The basis vectors define the axes and units for a coordinate plane
- The standard basis vectors are orthogonal to each other, and are equal, but they do not have to be.
