A basis of a vector space $V$ is a minimal spanning set - a set of linearly independent vectors that span $V$. A standard basis in $\R^n$, for example, is just

$$ \left\{\begin{pmatrix}1\\0\\0\\\vdots\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\\vdots\\0\end{pmatrix},\dots,\begin{pmatrix}0\\0\\0\\\vdots\\1\end{pmatrix}\right\} $$

As we can see, each vector is just a single direction in each “dimension”. In two and three dimensions, it is so common that we denote the unit vectors in each direction as $\hat{i}$, $\hat{j}$ and $\hat{k}$.

In fact, the dimension of a vector space is defined as the size of the basis.

For two vector spaces $V$ and $W$, it is true that

$$ \text{dim}(V) + \text{dim}(W) = \text{dim}(V+W) + \text{dim}(V \cap W) $$

<aside> 💡 Why is this true? Think about it like this: If $V$ has $v$ degrees of freedom, and $W$ has $w$. Say that there are no degrees of freedom in common, then when we combine the vector spaces the number of degrees of freedom of $V+W$ will be $v+w$, and $V\cap W$ won’t exist. However, if some degrees of freedom in $W$ are found in $V$ - variables are shared, for example - for each shared variable, the number of degrees of freedom in $V+W$ goes down by one, as there are fewer overall variables. Additionally, the number of degrees of freedom in $V \cap W$ will increase by one, as they now share an extra degree of freedom.

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