Is it true that vector spaces are defined to check if system of linear equations is solvable or not?
Explanation: Goal is to solve system of linear equations.
In matrix form: $Ax = b$. As $A = [C_1 \; C_2 \; ... \; C_n]$, where $C_n$ is a column and $x = [x_1 \; x_2 \; ... \; x_n]$.
Therefore, $C_1x_1 + C_2x_2 + ....+ C_nx_n = b$. Linear combination of column vectors produce vector $b$.
Because of above statement (linear combination) we choose a set of vectors that have closure under addition and scalar multiplication (closure under linear combination) and call that set of vectors a vector space. Now, if vector $b$ lies in that set of vectors (vector space) then only system of linear equations is solvable.