The problem is taken from Gathmann 2.33, from this link
Let $X$ be the set of all $2 \times 3$ matrices over a field $K$ that have rank at most $1$, considered as a subset of $\mathbb{A}^6 = Mat(2\times 3,K)$. Show that $X$ is an irreducible affine variety. What is its dimension?
There are several threads for this, but their approach is different than mine (invoking projective variates), and due to the location of this problem in the book, I am looking for a basic solution.
I have solved the irreducibility part. However, I am not sure about the dimension proof (I am sure that it is $4$, though).
It is easy to see that it is equal to finding the dimension of $Z(ae-bd,bf-ce,af-cd) \subset \mathbb{A}^6$. My strategy is to prove that $k[a,b,c,d,e,f]/\left<ae-bd,bf-ce,af-cd\right> \cong k[p,q,r,s]$, which then the dimension is obviously $4$. The irreducibility part shows that $\left<ae-bd,bf-ce,af-cd\right>$ is prime.
To do this, I construct a homomorphism $\phi' : k[a,b,c,d,e,f]/ \rightarrow k[p,q,r,s]$ that sends $a$ to $p$, $b$ to $q$, $c$ to $r$, $d$ to $sp$, $e$ to $sq$ and $f$ to $sr$. Because $ae-bd,bf-ce,af-cd$ are all in the kernel of this homomorphism by a direct check, this induces a homomorphism $\phi : k[a,b,c,d,e,f]/\left<ae-bd,bf-ce,af-cd\right> \rightarrow k[p,q,r,s]$ so that if $q$ is the quotient projection, $\phi \circ q = \phi'$. Also, consider the homomorphism $\tau' : k[p,q,r,s]/ \rightarrow k[a,b,c,d,e,f]$ that sends $p$ to $a$, $q$ to $b$, $r$ to $c$ and $s$ to $1$. Then, $\tau = q \circ \tau'$ is a homomorphism between $k[p,q,r,s]$ and $k[a,b,c,d,e,f]/\left<ae-bd,bf-ce,af-cd\right>$. By universal property, they must be inverses of each other, hence they are isomorphic. We conclude that the dimension is equal to $k[p,q,r,s]$, which is just 4.
Please comment on potential flaws/mistakes, and suggest a direction to correctness. Also, I would like some insight on understanding the intuition of dimension and connecting it to Krulls's dimension. Several sample problems similar with this question are also appreciated. Have a nice day!