I have recently been practicing the regular value theorem and I have asked to determine whether the set of upper triangular matrices is a submanifold for a 2 by 2 matrix represented below. $$ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix} iff (a,b,c,d) $$ my initial thoughts were to try and show that 0 is a regular value and since c=0 and calulating its derivative which is $$ \begin{pmatrix} 1 & 0 &0 &0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\ \end{pmatrix} $$ but this isn't surjective since we have a 0 row meaning it cannot be a submanifold. is this completely wrong
2026-03-30 05:29:00.1774848540
is the set of 2 by 2 upper triangular matrices an submanifold?
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