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nulliq.dev | ||
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austinmorlan.com
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| | | | | It took me longer than necessary to understand how a rotation transform matrix rotates a vector through three-dimensional space. Not because its a difficult concept but because it is often poorly explained in textbooks. Even the most explanatory book might derive the matrix for a rotation around one axis (e.g., x) but then present the other two matrices without showing their derivation. Ill explain my own understanding of their derivation in hopes that it will enlighten others that didnt catch on right a... | |
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codethrasher.com
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| | | | | A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar) \begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation} Simplistically, covectors can be thought of as "row vectors", or: \begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation} This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. | |
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www.paepper.com
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| | | | | As a data scientist, you are dealing a lot with linear algebra and in particular the multiplication of matrices. Important properties of a matrix are its eigenvalues and corresponding eigenvectors. So let's explore those a bit to get a better intuition of what they tell you about the transformation. We will just need numpy and a plotting library and create a set of points that make up a rectangle (5 points, so they are visually connected in the plot): | |
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mathscholar.org
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| | | [AI summary] The text presents a detailed, self-contained proof of the Fundamental Theorem of Calculus (FTC) using basic principles of calculus and real analysis. It breaks the proof into two parts: Part 1 establishes that the integral of a continuous function defines a differentiable function whose derivative is the original function, and Part 2 shows that the definite integral of a continuous function can be computed as the difference of an antiderivative evaluated at the endpoints. The proof relies on lemmas about continuity, differentiability, and the properties of integrals, avoiding advanced techniques. The text is structured to provide a clear, step-by-step derivation of the FTC for readers familiar with calculus fundamentals. | ||
