Basis of Vector Spaces

What is it?

In Linear Algebra, the basis of a vector space $\mathbf{V}$ consists on the smallest number of vectors in $\mathbf{V}$, which are the generators of $\mathbf{V}$. In other words, the basis of $\mathbf{V}$ is a set of vectors $\mathbf{S}$, if $\mathbf{S}$ is the generator of $\mathbf{V}$ and is linearly independent.

Dimension of vector spaces

The dimension of a not null vector space $\mathbf{V}$, corresponds to the number of vectors of a basis of $\mathbf{V}$.

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Canonical Bases

A canonical basis, also known as standard basis, uses of canonical vectors, which are “one-hot encoded vectors”. For example, the standard, or canonical basis of ${\mathbf{R}}^4$ would be:

$$\begin{array}{rl}& S_{{\mathbf{R}}^4}=\{(1,0,0,0),\\ & (0,1,0,0),\\ & (0,0,1,0),\\ & (0,0,0,1)\}\end{array}$$

We can also apply the same concept for Matrices. A standard basis for a vector space ${\mathbf{M}}_2$, based on matrices of order $2\times 2$, would be:

$${\mathbf{S}}_{{\mathbf{M}}_2}=\{\left|\begin{array}{cc}1& 0\\ 0& 0\end{array}\right|,\left|\begin{array}{cc}0& 1\\ 0& 0\end{array}\right|,\left|\begin{array}{cc}0& 0\\ 1& 0\end{array}\right|,\left|\begin{array}{cc}0& 0\\ 0& 1\end{array}\right|\}$$

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Checking the basis of a vector space

Given $\mathbf{S}=\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$, with ${\mathbf{v}}_1=(1,2,0)$, ${\mathbf{v}}_2=(0,4,1)$, and ${\mathbf{v}}_3=(2,0,2)$, we can check if these vectors are the basis of ${\mathbf{R}}^3$, by proving they are linearly independent, and that they generate ${\mathbf{R}}^3$.