$Sum of column spaces$

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$Sum of column spaces$

Let $K$ be a field and $C$ a nonzero $m \times n$ matrix over $K$. Show $\sum_{P \in \operatorname{GL}_m(K)}\operatorname{Im}PC=K^m$, where $\operatorname{Im}$ denotes the column space.

Linear Algebra

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Let the columns of $C$ be $C_1, \ldots C_n$ which may be regarded as elements of $K^m$. Since $C$ is nonzero, some $C_i$ is nonzero.

Because $\rm{GL}_m(K)$ has a transitive group action on $K^m$, there exists maps $P_1, \ldots P_m \in \rm{GL}_m(K)$ such that $P_jC_i = e_j$. Thus the image of $P_jC_i$ contains the span of $e_j$. Hence, these images span all of $K^m$.

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$Sum of column spaces$

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