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The following is a linear algebra problem I have been stuck for a while now.

Be $F_i=(a_{i1}, a_{i2}, \dots, a_{im}) \in M_{1\times n}(K)$ for $i=1,2,\dots,m.$ Let $W$ be the vector subespace of $M_{1\times n}(K)$ generated by $F_1,F_2,\dots F_m$. Let $A=(a_{ij}) \in M_{m \times n}(K).$ With Gauss-Jordan elimination using only row operations we convert the matrix $A$ to the matrix $A'=(a'_{ij})$ with exactly $r$ rows not full of zeros.

Prove that the $r$ rows not full of zeros of $A'$ form a basis of $W$.

I know these $r$ rows are linearly independent, but how can they form a basis of $W$ when $W$ is generated by $m$ vectors and $ r\leq m$?

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  • $\begingroup$ First of all, you probably mean that $F_i = ( a_{i1}, a_{i2}, \ldots, a_{in}).$ Second of all, apparently the set $\{F_1, \ldots, F_m\}$ is not a set of linearly independent vectors. $\endgroup$ – thanasissdr Jan 24 '17 at 5:01

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