While studying for Linear Algebra, I stumpled upon two questions, both formatted the same, yet one being more difficult for me than the other.
Question A Given a finite dimensional linear subspace $V$ with it's basis $\{v_1, \cdots, v_n\}, n \in \mathbb{N} $, and a linear subspace $W$ consisting out of the vectors $w_1, \cdots, w_n$ Proof that there exists are Linear Transformation $L$, such that $L(v_i) = w_i$, for $i = 1, 2, \cdots, n$.
My Answer Because the basis of $V$ is given by $v_1, \cdots, v_n$ we can also represent it as $\mathbb{I}_n$ in matrixform. We can describe any linear transformation with a matrix multiplication. Let's describe $L$ with $L_{mat}$ If we describe al vectors $w_1, \cdots, w_n$ in matrix $W_{mat}$, the following equation will hold:
$L_{mat} \cdot \mathbb{I}_n = W_{mat}$, therefore: $L_{mat} = W_{mat} * \mathbb{I}_n^{-1}$, so $L_{mat} = W_{mat}$. Because $W_{mat}$ exists, $L_{mat}$ will too.
Question B Assuming this proof is correct, how to prove this for infinite sized linear subspaces $V$ with basis $\{v_i \mid v_i \in \mathbb{N}\}$ and subspace $W$ with vectors $w_i, i \in \mathbb{N}$? I don't see what is different now? What can or can't I do? I know that infinite linear subspaces don't really have a basis, or it's at least hard to define them. Therefore it's also nearly impossible to express them in a matrix. How to approach this problem?