The fee/change address of cryptograffiti.info.
Given a linear operator over a space that has a inner product defined, we define the adjoint operator (the symbol is called "dagger") as the unique operator that satisfies:
Linear map of two variables.
More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from:that is linear on the first two arguments from X and Y, i.e.:Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.
The most important example by far is the dot product from , which is more specifically also a symmetric bilinear form.
Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g. .
Some definitions require both of the input spaces to be the same, e.g. , but it doesn't make much different in general.
The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.
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Effect of a change of basis on the matrix of a bilinear form by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16If is the change of basis matrix, then the matrix representation of a bilinear form that looked like:then the matrix in the new basis is:Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.
Proof: the value of a given bilinear form cannot change due to a change of basis, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:and in the new basis:and so since:
The first transaction of each Bitcoin block is called the "coinbase transaction", and it is magic as it does not need to point to a previous output script and have a valid input script as it serves as a Block reward for miners.
See form.
Analogous to a linear form, a multilinear form is a Multilinear map where the image is the underlying field of the vector space, e.g. .
The input script of the Coinbase transaction can be anything, and this can be used as a Bitcoin inscription method.
Notable examples:
- Genesis block message
- Prayer side of the Prayer wars
Subcase of symmetric multilinear map:
The most important example is the dot product, which is also a positive definite symmetric bilinear form.
Tested on Ubuntu 23.10 with P14s:TODO fails with:
sudo apt install hipcc
git clone https://github.com/ROCm/HIP-Examples
cd HIP-Examples/HIP-Examples-Applications/HelloWorld
make/bin/hipcc -g -c -o HelloWorld.o HelloWorld.cpp
clang: error: cannot find ROCm device library for gfx1103; provide its path via '--rocm-path' or '--rocm-device-lib-path', or pass '-nogpulib' to build without ROCm device library
make: *** [<builtin>: HelloWorld.o] Error 1The prototypical example of it is the complex dot product.
Note that this form is neither strictly symmetric, it satisfies:where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.
Bibliography:
Pinned article: Introduction to the OurBigBook Project
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Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
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Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
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Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
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