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**Joanna** $\displaystyle b_{1}, b_{2},...,b_{r}$ are a basis for the rowspace of A, so every combination of the rows of A can be written in terms of this basis.

Each of $\displaystyle v_1, v_2, ...\,\,\, and\,\,\, v_m$ is in the row space of A, and so can be written in terms of $\displaystyle b_{1}, b_{2},...,b_{r}$.

Did you understand column spaces? Row space works the same way, except obviously related to rows rather than columns. If you did understand column space,

note that the row space of $\displaystyle A$ is equal to the column space of $\displaystyle A^{T}$, and a basis for the row space of $\displaystyle A$ is a basis for the column space of $\displaystyle A^{T}$