For $A=[a_1\ \cdots\ a_n]\in\mathbb F^{m\times n}$ and $x=(x_1,\ldots,x_n)^T$, define $Ax=x_1a_1+\cdots+x_na_n$. Equivalently, $(Ax)_i=\sum_j a_{ij}x_j$. Thus $Ax$ is a vector in $\mathbb F^m$ and is a linear combination of the columns of $A$.
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