Answer to Question #24236 in Linear Algebra for Adam J
Do row operations preserve the linear independence among the columns of a matrix? How about the rows of a matrix?
The columns of a matrix are linearly independent if and only if the column rank of the matrix equals the number of columns. It is well-known that the column rank of a matrix coincides with its row rank. Therefore, it is sufficient to prove that a row operation preserves the (row or column) rank.
Each elementary row operation can be regarded as a multiplication by the matrix of this operation. It can be easily seen that this matrix is invertible (simply because each elementary row operation is obviously invertible). We know that multiplication by an invertible matrix does not change the rank of the initial matrix. Thus, a row operation preserves the rank. Answer:& row operations preserve the linear independence of rows as well as the linear independence of columns.