# All Questions

14k views

### “Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
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### Bijection between Cartesian products.

Problem: Show that there is a bijective correspondence of $A \times B$ with $B \times A$. Here is my proof. I want to see if I'm correct about the injective portion of my proof and some ...
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### Showing that $(A_{ij})=\left(\frac1{1-x_ix_j}\right)$ is positive semidefinite

Consider the matrix $A$ whose elements are $A_{ij} = \frac{1}{1-x_i x_j}$ where we have $-1 < x_i < 1$ and $-1 < x_j < 1$ for $i,j=1,2,...n$. For example, when $n=3$ the matrix ...
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### Is $\dim\left(U+V\right)=\dim\left(U\cup V\right)$?

If $U$ and $V$ are proper subspaces of a vector space $W$, is $\dim\left(U+V\right)=\dim\left(U\cup V\right)$?
If a measure $P$ can take value of $+\infty$, does countable additivity ($\sigma$-additivity) imply continuity at $\emptyset$? If not, what is a good example?