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Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ be a linear surjective map with $n \geq m$, if we denote by $A$ the matrix of $F$,is it true that the rank of $A$ is always $\neq 0$? Why?What can we say about the columns or lines of $A$?


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Rank is the dimension of the range. If the map is surjective, what is the range? What is the dimension of the codomain? –  Bill Cook Feb 14 '12 at 22:08

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1) $A$ is the $m\times n$ matrix such that $F(x)=A(x)$.

2) The range of $F$ is the column space of $A$.

3) The rank of $A$ is the dimension of its row space.

4) For any matrix $A$, the rank of $A$ is equal to the dimension of the column space of $A$.

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