⧼exchistory⧽
5 exercise(s) shown, 0 hidden
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Work out the geometric interpretation of the map [math]f(x)=Ax[/math], with
[[math]]
A\in M_2(\pm1)
[[/math]]
and then discuss as well the diagonalization of these matrices.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Diagonalize explicitely the third flat matrix, namely
[[math]]
\mathbb I_3
=\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}
[[/math]]
and then study as well the general case, that of the matrix [math]\mathbb I_N[/math].
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Work out the trigonometry formulae
[[math]]
\sin(2t)=2\sin t\cos t\quad,\quad
\cos(2t)=2\cos^2t-1
[[/math]]
by using elementary methods, coming from plane geometry.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Prove that the isometries in [math]2[/math] dimensions are either rotations, or symmetries, as to complete the proof of Theorem 1.40.
BBot
Apr 20'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
Develop a theory of angles between the vectors [math]x,y\in\mathbb R^N[/math], by using the well-known formula
[[math]]
\lt x,y \gt =||x||\cdot||y||\cdot\cos t
[[/math]]
that you should by the way fully understand first, in [math]N=2[/math] dimensions.