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Given a solution set of a system of linear equations with dim(solution_set) = dim(SolSet) = 1, could it be the Image of a linear equation?

What we know is that the solution set is equal to the Kernel of a linear transformation, right?

So using the rank-nullity theorem we get this:

$ \dim V = \dim\mathrm{Im}f + \dim\mathrm{Ker}f$
($\dim\mathrm{Ker}f = \dim(\text{SolSet})$ )

$ \dim V = \dim\mathrm{Im}f + \dim(\text{SolSet}) \Rightarrow \dim\mathrm{Im}f = \dim V - \dim(\text{SolSet}) $

For the $\dim\mathrm{Im}f$ to be equal to the of $\dim(\text{SolSet})$, $\dim V$ must be 2 times $\dim(\text{SolSet})$. (maybe using the 1st isomorphism theorem?) Is that possible?

Thank you for your time!

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    The solution set of a system of linear equations is the image of a linear transformation if and only if it is a subspace, if and only if the system of equations is homogeneous.2012-04-30

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