⧼exchistory⧽
4 exercise(s) shown, 0 hidden
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Fill in all the geometric details in the basic theory of the determinant, by using the same type of arguments as those in the proof of
[[math]]
\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc
[[/math]]
which was fully proved in the above, namely geometric manipulations, and Thales.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Prove with full details, based on the above, that the determinant of the systems of vectors
[[math]]
\det:\mathbb R^N\times\ldots\times\mathbb R^N\to\mathbb R
[[/math]]
is multilinear, alternate and unital, and unique with these properties. Then try to prove as well this directly, without any reference to geometry.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Work out, with full details, the theory of the signature map
[[math]]
\varepsilon:S_N\to\{\pm1\}
[[/math]]
as outlined in Theorem 2.35 and its proof.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Prove that for a matrix [math]H\in M_N(\pm1)[/math], we have
[[math]]
|\det H|\leq N^{N/2}
[[/math]]
and then find the maximizers of [math]|\det H|[/math], at small values of [math]N[/math].