Answer
See below
Work Step by Step
Given $V = R^3$, $S$ is the subspace consisting of all points lying on the plane with Cartesian equation
$$x+4y-3z=0$$
We can rewrite as $x=3z-4y$
Since $dim (A)=dim (R^3) \rightarrow dim(D)=2$
Let take vectors $(1,1,0)$ and $(0,0,1)$ in $S$
then we obtain: $v_1=(3.1-4.0,0,1)=(3,0,1)\\
v_2=(3.0-4.1,1,0)=(-4,1,0)$
Since $v_1,v_2$ are already in $S$, then $v_3=(1,0,0)$
Hence, $(3,0,1),(-4,1,0),(1,0,0)$ is a base for $R^3$
