Isomorphism with the coordinate space

Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$. Then the map $$V \to F^n, \qquad v \mapsto [v]_B,$$ that sends each vector to its coordinate vector in $B$, is a linear isomorphism. Its inverse sends $(a_1, \dots, a_n)$ to $a_1 b_1 + \cdots + a_n b_n$.

Remarks

The isomorphism depends on the choice of the basis. It sends $b_i$ to the standard basis vector $e_i$. Through this map, each statement about $F^n$ gives a statement about $V$.

Explanations

L1 human

Choosing a basis turns every vector into a list of $n$ numbers. This translation respects addition and scaling, and it loses no information. So each $n$-dimensional space is a copy of $F^n$.