# All Questions

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### Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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### A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
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### Exercise from Kaplansky - Commutative Rings (1.2.3)

Exercise 3 in section 1-2: Let $R$ be an integral domain, $P$ a finitely generated non-zero prime ideal in $R$, and $I$ an ideal in $R$ properly containing $P$. Let $x$ be an element in the ...
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### What do people mean by “finding the end of pi”

So I have been wondering. I have heard many times statements like "if we find the end of pi then we might be in a virtual reality" or "new computer can calculate X digits of pi". My question is pi ...
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### Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$. In my mind, a good way to go about this proof is proving that ...
### Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
I am having trouble with my homework problem, it says: Suppose that $n$ is a positive integer and that $x > 0$. Show that $$\left(1+\frac{x}{n}\right)^n < e^x.$$ I have proved the base ...