Finding basis of a column space/row space

[ashina14]

I was wondering whether we can use row as well as column operations to reduce a matrix to find column space? Or do we only have to perform row operations to reduce matrix in case of row space and column operations to find column space?

 

[HallsofIvy]

You could, of course, take the transpose of a matrix, so that "columns" become rows and vice versa, but the "column space" of a matrix (the space spanned by its columns as vectors) is NOT, in general, the same as the "row space" of a matrix (the space spanned by it rows). If the matrix is not square, the will have completely different dimensions. If the matrix is square, the row space and column space will have the same dimension and so be "isomorphic" but not, in general, the "same" spaces.