For the following matrix:
$$ \begin{bmatrix} 1 & 2 & 1 & 3 \\ 2 & 5 & 5 & 6 \\ 3 & 7 & 6 & 11 \\ 1 & 5 & 10 & 8 \\ \end{bmatrix} $$
I want to find a basis for the Row space, Column space, Kernel (if it represents a Linear Transformation), and Image(if it represents a Linear Transformation).
The row echelon form of this matrix is: $$ \begin{bmatrix} 1 & 2 & 1 & 3 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $$
Row space basis:
$$\{(1,2,1,3),(0,1,3,0),(0,0,0,2)\}$$
Column space basis:
$$\{(1,2,3,1),(2,5,7,5),(3,6,11,8)\}$$
Kernel basis:
$x+2y+z+3w=0$
$y+3z=0$
$2w=0$
$(x,y,z,w)=z(5,-3,1,0)$
$$\{(5,-3,1,0)\}$$
Image basis (same as column space):
$$\{(1,2,3,1),(2,5,7,5),(3,6,11,8)\}$$
Are my above solutions correct? Of not then why?