Let $A$ be a $5\times 4$ matrix with real entries such that the space of all solutions of the linear system $AX^t = (1,2,3,4,5)^t$ is given by$\{(1+2s, 2+3s, 3+4s, 4+5s)^t :s\in \mathbb{R}\}$ where $t$ denotes the transpose of a matrix. Then what would be the rank of $A$?
Here is my attempt
Number of linearly independent solution of a non homogeneous system of linear equations is given by $n-r+1$ where n refers to number of unknowns and $r$ denotes rank of the coefficient matrix $A$. Based on this fact, we may write $n-r+1 =1 $. Since there seems only one linearly independent solution (Here i am confused). Am i right? or How can i do it right? thanks