Let $f, g, h \in L$, where L is the vector space of all linear maps that map from $\mathbb{R}^3 \rightarrow \mathbb{R}^2$.
$ f \left(\left( \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right) \right) = \left( \begin{array}{cc} x_1 + x_2 + x_3 \\ x_1 + x_2 \\ \end{array} \right)$
$ g \left(\left( \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right) \right) = \left( \begin{array}{cc} 2x_1 + x_3 \\ x_1 + x_2 \\ \end{array} \right)$
$ h \left(\left( \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right) \right) = \left( \begin{array}{cc} 2x_2 \\ x_1 \\ \end{array} \right)$
Show that f, g and h are linearly independent.
Could I show this by using the matrices of those linear maps, which are uniquely determined, and then show that there aren't two of those matrices that are equal, proving that there isn't a linear combination of f, g and h, besides the trivial one, that is equal to 0?