Change of Coordinates

In many applications, we use multiple coordinate systems to describe locations or positions in 2-D or 3-D. In aerospace engineering, we can describe a position using earth-fixed coordinates or body-fixed coordinates, where the body refers to an aircraft. See more here. Earth-fixed coordinates are with respect to a specific origin, with the three axes point East, North, and straight up. The origin of the body-fixed coordinates is a specific location on the aircraft (typically the center of gravity), and the axes point forward, left, and up. …

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Geometric Transformations

Corrolary. I would have liked to have graphs along with this post. It would have been illustrative. Yet, Matplotlibb is an absolute mess. I dislike it a lot), and I couldn’t stomach wasting a whole day (or more) trying to get something as basic as plotting vectors to work. It’s infuriating. import numpy as np Suppose the 2-vector $$x$$ represents a position in 2-D space. Several important transformations or mappings from points to points can be expressed as matrix-vector products $$y = Ax$$, with $$A$$ a $$2 \times 2$$ matrix. …

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Data-Drivel

Every time someone utters the words data-driven, data-smart, data-informed, and similar buzzwords, my brain goes to this. This might sound like gatekeeping, but humor me for a moment. Having a company culture orbiting around dashboards and reports is not the same as being data-driven. If people disagree with the numbers, it could mean that they’re not aligned about the context of said numbers. You can have “inaccurate” (and incomplete) data that is still directionally true and valuable. …

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If $$A$$ is a $$m \times n$$ matrix, its transponse, $$A^T$$, is the $$n \times m$$ matrix given by $$(A^T)_{ij} = A_{ji}$$. In words, the words and columns and transposed in $$A^T$$. If we transpose a matrix twice we get the original matrix, $$(A^T)^T = A$$. import numpy as np A = np.array([[0, 4], [7, 0], [3, 1]]) A ## array([[0, 4], ## [7, 0], ## [3, 1]]) A.transpose() ## array([[0, 7, 3], ## [4, 0, 1]]) A. …
Zero Matrix. A zero matrix is a matrix with all elements equal to zero. A zero matrix of size $$m /times x$$ is sometimes written as $$0_{m \times n}$$, but usually a zero matrix is denoted just 0. import numpy as np np.zeros((2, 2)) ## array([[0., 0.], ## [0., 0.]]) Identity Matrix. An identity matrix is another common matrix. It is always square. It’s diagonal elements are all equal to one, and it’s off-diagonal elements are zero. …