Matrix/Linear system/Solution set and preimage/Exercise

Let ${}M$ be an ${}m\times n$-matrix over the field ${}K$, let ${}\varphi \colon K^{n}\rightarrow K^{m}$ be the corresponding linear mapping, and let ${}Mx=c$ denote (depending on a vector ${}c\in K^{m}$) the corresponding system of linear equations. Show that the solution set of the system equals the preimage

of ${}c$ under the linear mapping ${}\varphi$.

Create a solution