Row Space using a Vector Representation

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 ?

• I don't fully understand your request. Yes, each way you have written the span so far is correct. I will point out that if you were to perform row reduction first, you can have a simpler expression, here being $\{(a,b,2b)~:~a,b\in\Bbb R\}$. I do not understand what you are referring to in the last paragraph except perhaps in reminding you that you can row reduce before representing in order to reduce potential redundancy and to possibly make the final representation cleaner. May 4, 2023 at 12:59
• @JMoravitz So you are saying reduce to reduced echelon form first ?
– user1071088
May 4, 2023 at 13:16

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 $$\operatorname{row}(N)=\{(x,y,z)\in\Bbb R^3\mid z=2y\}.$$
• I think that their phrase "generic vector" means nothing more than "writing in the form row(N) = {($x,y,z$) $\in \mathbb R^3$ | equation(s)}". May 4, 2023 at 13:18
• Yes. And the fact that there is only one equation could be foreseen: the subspace is of codimension $3-2=1.$ May 4, 2023 at 13:21
• Yes, I read it at the beginning of your post. These are the 2 sentences I was talking about. But maybe @JMoravitz's guess was right and the second sentence asks you to write $\operatorname{row}(N)=\{(x,y,2y)\mid x,y\in\Bbb R\}.$ May 4, 2023 at 13:28