Row operations do not affect dependence between columns of a matrix

• The theorem in reference basically says that a/any linear combination of the columns of $A$ can be equal to 0 iff a/any linear combinations of the columns of $B$ can be equal to 0.
• Where $B$ is obtained through row operations on $A$.
• Since equating linear combinations of vectors to zero helps us test linear dependence of the set [1], the theorem says that if the columns were dependent before the row operations, the columns will still be dependent.