For the matrix N = $$\begin{pmatrix} 1& 1& 2 \\ 0& 1& 2\\ \end{pmatrix}$$
determine the row space of N, writing in the form row(N) = {($x,y,z$) $\in \mathbb R^3$ |...}. Hence, write row(N) using a generic vector.
I know that N is in echelon form. I can write row(N) = span{$(1,1,2),(0,1,2)$}. I want to write in the form given, so I wrote row(N) = {($a,a+b,2a+2b$) $\in \mathbb R^3$ | $a,b \in \mathbb R$}.
I think this is correct ?
But I am unsure on how to write it using a generic vector. If it said generic vector$\mathbf{s}$, I would just write row(N) = span{$(1,1,2),(0,1,2)$}. These two vectors are linearly independent so I don't know what more I can do ?
Your "row(N) = {($a,a+b,2a+2b$) $\in \mathbb R^3$ | $a,b \in \mathbb R$}" is correct, but "writing in the form row(N) = {($x,y,z$) $\in \mathbb R^3$ |...}" asks you to find an equation satisfied by $(x,y,z)$ iff there exists $a,b\in\Bbb R$ such that $$x=a,\quad y=a+b,\quad z=2a+2b.$$ For this, you have to eliminate the parameters $a,b.$ The equations above are equivalent to $$a=x,\quad b=y-x,\quad z=2y,$$ hence $$\operatorname{row}(N)=\{(x,y,z)\in\Bbb R^3\mid z=2y\}.$$