Assume that I have a $3 \times 3$ matrix $A$ with columns $A_1$, $A_2$, and $A_3$ that are linearly independent. Say that I want to find the column space of A. Isn't it possible for me to find some combination of $A_1$, $A_2$, and $A_3$ such that I can come up with $ \left[ {\begin{array}{c} 1 \\ 0 \\ 0 \end{array} } \right]$, $\left[ {\begin{array}{c} 0 \\ 1 \\ 0 \end{array} } \right]$ and $\left[ {\begin{array}{c} 0 \\ 0 \\ 1 \end{array} } \right]$ and just say that the columns of $A$ span all of $\mathbb{R}^3$?
Is anything stopping me from doing this? What is it that limits column spaces?