Answer
False
Work Step by Step
Let $S=\{(1,0);(0,1);(0,0)\}$ be a spanning set for a vector space $V=R^2$
Obtain $(x,y)=x(1,0)+y(0,1)+(0,0), \forall x,y \in R^2$
so $S'=\{(1,0);(0,1)\}$ is a spanning set for $R^2$
Hence, the statement is false.