Given a matrix $A:$
\begin{pmatrix} 1 & 2 & 5 \\ 3 & 5 & 13 \\ -2 & -1 & -4 \end{pmatrix}
My textbook has reduced it to RREF to find kernel and dimension of it. To find the basis for image, they have taken $A^{T}$ and then reduce it to RREF. I don't understand why this has been done; this is my main point of frustration.
Also, let me know if there are any alternative ways to find the basis for the image. Thank you in advance.
Best Answer
If you have the RREF, you can pick the columns of $A$ corresponding to the pivot columns of the RREF.
If you want a simplified basis, you cannot use the RREF of $A$, as row operations will change the column space. Instead, your book does column operations on $A$ (row operation on $A^T$) to eliminate the columns of $A$ down to echelon form.