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Let $f \colon \mathbb{C}^5 \rightarrow \mathbb{C}^7$ a linear function, $f(2 i e_1 + e_3) = f(e_2)$ and $\mathbb{C}^7=X \oplus Im(f)$. What dimension has $X$?

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    I'm not French. Thank you for the correction2011-05-24

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Hint: Apply the rank-nullity theorem. Given that $f$ satisfies at least the relation that you listed, what are the possible dimensions of the kernel?

Second hint: Take a basis $v_1, \ldots v_5$ of $\mathbb{C}^5$ such that $v_1=2ie_1-e_2+3_3$

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    @Katy23 They are, but to explain why, you should look into the following strengthening of the rank-nullity theorem. If $V'\subset V$ and $W'\subset W$ are vector spaces with $\dim V'+\dim W'=\dim V$, then there is a linear map $f:V\to W$ with $\ker f = V'$ and $\operatorname{im} f = W'$, which is the composition of $V\to V/V' \to W' \to W$, where the first map is the natural quotient, the last is the natural inclusion, and the middle map is an isomorphism. It is easy to see this by taking a basis in the right way.2011-05-24