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The column space is the span of the columns, and its dimension is the rank. Row reduction identifies which columns are pivots; the corresponding original columns form a basis, not the reduced ones.
Column space
Col(A) = span of the columns; dim Col(A) = rank(A)
Row operations change the column space, though they preserve which columns are independent — so the pivot positions transfer but the entries do not.
Yes, always. That equality is one of the fundamental results of linear algebra.