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Questions about the properties of vector spaces and linear transformations, including linear systems in general.
15
votes
Accepted
Endomorphisms of the p-adic group $(\mathbb Z_p,+)$
I claim that any endomorphism of $\mathbb{Z}_p$ is just multiplication by a constant.
Let $f$ be an endomorphism of $\mathbb{Z}_p$. First observe that $p^n f(x) = f(p^n x)$, so $f$ takes $p^n \mathbb …
10
votes
Accepted
Deceptive linear algebra problem
Let $$f(X) = (X - a_1) \cdots (X - a_p) = X^p + e_1 X^{p-1} + \dots + e_p.$$ Using $f(a_i) = 0$ for each $i$, we get
\begin{equation} \begin{bmatrix} d_{p+1} & d_p & \cdots & d_1 \\ d_{p+2} & d_{p+1} …