Answer
False
Work Step by Step
Let $m=2$ and $n=3$
and the spanning set for $R^2$ is $\{(1,0),(0,1),(1,1),(2,2)\}$. This set has four vectors.
Then, the spanning set for $R^3$ could be $\{(1,0,0),(0,1,0),(0,0,1)\}$. This set has only three vectors.
Hence, the statement is false.
