Consider two vectors
$$
u_{1} = \begin{bmatrix}
-1 \\
3 \\
2 \\
\end{bmatrix},\quad
u_{2} = \begin{bmatrix}
6 \\
1 \\
1 \\
\end{bmatrix}
$$
I can tell they don't span $R^3$ because $R^3$ requires three vectors to span it.
But is there another way I can see that it does not span $R^3$, for example, in terms of overdetermined system, or not a square matrix? Just trying to develop a mature understanding of this topic, that's all.
Best Answer
You want to find $(a,b,c)$ such that $(a,b,c)$ cannot be written as $x(-1,3,2)+y(6,1,1)$. But this is easy -- in fact, since most $(a,b,c)$ cannot be written in that way, you can just pick one at random and start solving!
For example, pick $(a,b,c)=(0,0,1)$.
Since $-x+6y=0$ we must have $x=6y$. Now plug that into $3x+y=0$, giving $18y+y=0$ which is the same as $19y=0$, and so $y=0$.
But then $x=y=0$ and $x(-1,3,3)+y(6,1,1)=(0,0,0)$, which is not what we wanted. So $(0,0,1)$ is impossible to make, and thus we have the counterexample we need.