A theorem concerning unique linear mapping between vector spaces: What does it say?

Theorem (from Schaum's Linear Algebra) Let $V$ and $U$ be vector spaces and $\{v_1, \ldots, v_n\}$ be a basis on $V$. Let $\{u_1,\ldots, u_n\}$ be arbitrary vectors in $U$. Then there exists a unique linear mapping $F: V \to U$ such that $F(v_i) = u_i$.

I omit the proof, it is not so hard. But I don't understand what this theorem means or what does it imply, how one can use it to deduce more results. Can anybody briefly explain it? Thanks in advance!

The theorem says that the values of a linear transformation $T:V\to U$ are completely and freely determined by the values $T$ attains on a basis of $V$. In more detail, if $T,S:V\to U$ are two linear transformations and $\{v_1 ,\cdots v_n\}$ is a basis for $V$ and $T(v_i)=S(v_i)$ holds for all $v_i$ then $S=T$ (which means that for any $v\in V$ holds that $T(v)=S(v)$). This is the precise meaning of $T$ is completely determined by its values on a basis.

Further, given any (free) choice of vectors $u_1 ,\cdots, u_n$ in $U$ there is a linear transformation $T:V\to U$ such that $T(v_i)=u_i$. This is the meaning of $T$ is freely determined by its values on a basis.

These properties are extremely useful when studying and constructing linear transformations since they reduce the study of an arbitrary linear transformations $T:V\to U$ to understanding what $T$ does on a basis. If $V$ has a finite basis then this process reduces considering the value of $T$ on infinitely many values to just finitely many.

For example, taking the indefinite integral of a function is a linear operator (since it respects addition of functions and scalar multiplication). If you consider the linear space $P$ of all polynomials then you obtain a linear transformation $\int :P\to P$. It is useful to know that the values of $\int$ are determined uniquely by its values on a basis, for instance the basis $\{1,x,x^2,x^3,\cdots\}$.

Another example considers the space $\mathbb R^2$. Given any two points $v_1,v_2$ it is convenient to know that there exists a linear transformation $T:\mathbb R^2\to \mathbb R^2$ with $T(e_1)=v_1$ and $T(e_2)=v_2$ since then you can study properties of these two values by studying properties of the linear transformation.