>[!summary] A basis is our lowest number of vectors to build coordinates in a space, and it changes the way you view vectors in a space. > **Requirements of a Basis:** >- All vectors in a set are linearly independent >- span the space (Have enough vectors to map a space) >[!info]+ Read Time **⏱ 2 mins** # Euclidean Definition A basis is the lens through which you view a space. It's a "recipe book" of the lowest number of vectors to build coordinates in a space. A basis changes the way you view vectors in a space. The requirements for a basis in $\mathbb{R}^{\textcolor{orange}{n}}$ are the following: - There are $\textcolor{orange}{n}$ amount of [[Vectors Addition & Multiplication|vectors]] in a [[Sets|set]] and $\textcolor{orange}{n}$ vectors are [[Linear Independence & Dependence|linearly independent]] - The set of vectors [[Span|span]] the space > [!info]- Logical argument for why these requirements must be true In order to create any possible vector in a space like $\mathbb{R^2}$ you would need vectors who are not [[Linear Combinations|linear combinations]] of other vectors in a set (Since this would fail at mapping any vector) > As well spanning the space, can be argued. In the case of $\mathbb{R^2}$ you would need two vectors to span that space, who are [[Linear Independence & Dependence|linearly independent]] (from the first requirement) ## Matrix Requirements A given matrix with columns $A=\left[ \vec{v_{1}},\dots,\vec{v_{k}} \right]$ whose columns are in $\mathbb{R}^n$. The set of vectors $\left\{ \vec{v_{1}},\dots,\vec{v_{k}} \right\}$ is a basis for $\mathbb{R}^n$ if the following conditions are true: - [[Rank|Rank]] a of of the matrix is equal to $n$ ($\text{Rank(A)}=n$) - There exist exactly $k$ amount of [[Vectors Addition & Multiplication|vectors]] in the set so that $n=k$ --- 📂 Want to see more structured notes like these? Help grow the project by [starring Math & Matter on GitHub](https://github.com/rajeevphysics/Obsidan-MathMatter). ---