Thanks for your replies, I found the answer.
Consider Page 9, 10 of
http://www.ms.uky.edu/~lee/amspekulin/r ... n_ax=b.pdf
for an example.
As a further example, for an m x n matrix, where m is strictly greater than n, and all entries are non-zero, then, in the reduced row echelon form, the last row will necessarily be zero.
So for a 3 x 2 matrix although the column vectors of the original matrix span part of R^3, the column vectors of the reduced row echelon form will only span part of R^2.
Hence the answer to the original question is "Not necessarily", since
[Row Operations may affect
the column space of a matrix