TITLE: Understanding how to find a basis for the row space/column space of some matrix A.
QUESTION [7 upvotes]: I just need some verification on finding the basis for column spaces and row spaces.
If I'm given a matrix A and asked to find a basis for the row space, is the following method correct?
-Reduce to row echelon form. The rows with leading 1's will be the basis vectors for the row space.
When looking for the basis of the column space (given some matrix A), is the following method correct?
-Reduce to row echelon form. The columns with leading 1's corresponding to the original matrix A will be the basis vectors for the column space. 
When looking for bases of row space/column space, there's no need in taking a transpose of the original matrix, right? I just reduce to row echelon and use the reduced matrix to get my basis vectors for the row space, and use the original matrix to correspond my reduced form columns with leading 1's to get the basis for my column space.

REPLY [2 votes]: Your procedure is correct, and I'd like to summary them:
Finding Row Space

Strategy: do Gaussian elimination on the matrix M; non-zero rows of reduced matrix M' form the set spanning row space. 
Reason: 1) elementary row operations don't change the row space of M, since row vectors are changed linearly by elementary row operations; 2)The non-zero rows in a reduced matrix form its row space, which can be proven $\sum_i c_i\vec{r_i}={0}\Rightarrow c_i=0,\forall i$, indicating these row vectors are independent to form the row space. 

Finding Column Space

Strategy: 1)You could do it by transpose. 2)Or, do Gaussian elimination, take the columns of $M$ corresponding to columns with leading entries in $M'$.
Reason for 2): $M'$ implies solutions where columns without leading entries are the linear combination of others, so we only need to take the columns with leading entries in $M$.