Find all values of $t$ for which the system of equations
$$\begin{array} 22x_1 + x_2 + 4x_3 + 3x_4 = 1\\ x_1 + 3x_2 + 2x_3 − x_4 = 3t\\ x_1 + x_2 + 2x_3 + x_4 = t^2 \end{array}$$
has a solution?
I was given a theorem, that system has a solution, when column vector of RHS lies in the subspace spanned by column vectors of LHS. If we take respective column vectors, and notice that third is a scalar multiple of the first column, we get three linearly independent vectors. What I don't get is, why should there be particular $t$'s, for which the system doesn't have a solution, as if we have three linearly independent (column) vectors, they should span $\mathbb{R^3}$, and thus we could find solution for any set of $t$'s.
I suppose I'm wrong, but where's the mistake, and how should I check then, which $t$'s suffice?