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a college algebra student asked me to prove to her that any line with slope \frac{-1}{m} is perpendicular to a line with slope m. in my infinite hubris, i confidently launched into the dot product proof. about halfway through, i realised that i'd forgotten that adding non-zero constants to your column vectors' y-terms completely ruins the dot product. i absolutely flubbed it and will suffer for eternity as penance. send me to linear algebra hell.
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Who knew finding the Fourier sine series of f(x) (defined on upper right corner of page) would enable one to evaluate the sum of an infinite series (1/1^2 + 1/3^2 + 1/5^2 + ...) in the end!? Amazing.
Orthogonality
Introduction In mathematics and physics, orthogonality is a key concept, not just confined to vectors in Euclidean space but also extended to more abstract concepts like functions. Two functions are said to be orthogonal if their inner product is zero. Inner Product of Functions In the context of function spaces, the inner product of two functions $f(x)$ and $g(x)$, over an interval $[a, b]$,…
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Introduction Today, in this post, we will learn about Orthogonality in Mathematics and Computer Science. Mathematics Orthogonality is an outline of the idea of perpendicularity to the linear algebra of bilinear forms. When B (u, v) = 0, two elements u and v of a vector space with bilinear form B are orthogonal. The vector ... Read more
Introduction Today, in this post, we will learn about Orthogonality in Mathematics and Computer Science. Mathematics Orthogonality is an outline of the idea of perpendicularity to the linear algebra of bilinear forms. When B (u, v) = 0, two elements u and v of a vector space with bilinear form B are orthogonal. The vector ... Read more
Activating the higher circuits.
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The Orthogonality Thesis, Intelligence, and Stupidity
Let’s block ads! (Why?) …Artificial Intelligence
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An observation about the geometry of Pauli x,y matrices
An observation about the geometry of Pauli x,y matrices
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Motivation
The conventional form for the Pauli matrices is
\begin{equation}\label{eqn:pauliMatrixXYgeometry:20} \begin{aligned} \sigma_x &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \\ \sigma_y &= \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \\ \sigma_z &= \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \end{aligned}. \…
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M.C. Escher Relativity, 1953
