= Solution
Write the columns of $A$ as $a_1,\ldots,a_n$ and the coordinates of the <vector> $x$ as $x_1,\ldots,x_n$. By the definition of <matrix multiplication>,
$$
\boxed{y=Ax=x_1a_1+\cdots+x_na_n\in\operatorname{span}\{a_1,\ldots,a_n\}=\operatorname{Col}(A).}
$$
Indeed, its $i$th coordinate is $\sum_{j=1}^n A_{ij}x_j$, exactly the $i$th coordinate of this <linear combination>. Since the <column space> contains every <linear combination> of the columns, the claim holds for every <vector> $x$.
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