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I know how to verify that a basis for R^3 spans R^3 -- you just form the equation

(u1, u2, u3) = c1v1 + c2v2 + c3v3 ,

equate corresponding components, and see whether the system has a unique solution.

But for verifying that a set of vectors spans a column space, you're not trying to see whether it spans all of R3 (or R2, or R4, or whatever), correct? You're trying to see whether it spans whatever the column space spans. So you can't just set the linear combination equal to u, right? So then how would you go about verifying that a basis spans a column space?

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Suppose you claim that $\{ v_1, v_2, \ldots, v_m \}$ is a basis for the column space.

You just have to verify that $u_1$, $u_2, \ldots, u_n$ can be expressed as linear combinations of $v_1$, $v_2$,$\ldots, v_m$ where $u_i$ are the columns of the matrix.

The column space is formed by linear combinations of $u_1, u_2,\ldots u_n$. If $x$ is in the column space, then $$x = \sum_{i=1}^n a_i u_i$$ for some $a_i$.

if $u_i= \sum_{j=1}^m b_{ij}v_i$,

$$x=\sum_{i=1}^na_i\sum_{j=1}^mb_{ij}v_j=\sum_{j=1}^m\sum_{i=1}^na_ib_{ij}v_j$$

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