Math and stuff

Proposition Find a parametrization for each of the following paths: The circle $C[1 + i, 1]$, oriented counter-clockwise. The line segment from $-1 - i$ from $2i$. The top half of the circle $C[0, 34]$, oriented clockwise. The rectangle with vertices $\pm 1 \pm 2i$, oriented counter-clockwise. The ellipse $\{...

Proposition If $\omega \in \Lambda^n(V)$ is the volume element determined by $T$ and $\mu$, and $w_1, \cdots, w_n \in V$, show that \[\begin{align*} \abs{\omega(w_1, \cdots, w_n)} = \sqrt{\det(g_{ij})}. \end{align*}\] where $g_{ij} = T(w_i, w_j)$. Solution Let $v_1, \cdots, v_n$ be an orthonormal basis such that $[v_1, \cdots, v_n] = \mu$....

Proposition If $f: V \rightarrow V$ is a linear transformation and $\dim V = n$, then $f^{*}: \Lambda^n(V) \rightarrow \Lambda^n(V)$ must be multiplication by some constant $c$. Show that $c = \det f$. Solution $\Lambda^n(V)$ is a 1-dimensional vector space. Since $f^*$ is a linear map, it must be multiplication...

Proposition Let $A$ be a ring $\ne 0$. Show that the set of prime ideals of $A$ has minimal elements with respect of inclusion. Solution Let $\Spec(A)$ denote the set of all prime ideals of $A$. By Theorem 1.3 on P.3 of Atiyah, $A$ contains at least one maximal ideal...

Proposition Find the absolute values of $-2i(3 + i)(2 + 4i)(1 + i)$ $\displaystyle\frac{(3 + 4i)(-1 + 2i)}{(-1 - i)(3 - i)}$. Solution $\abs{-2i(3 + i)(2 + 4i)(1 + i)} = 2 \cdot \sqrt{10} \cdot \sqrt{20} \cdot \sqrt{2} = 40$. $\frac{5 \cdot \sqrt{1 + 4}}{\sqrt{2} \cdot \sqrt{9 + 1}} =...