Tag: math

Orthogonal projections in linear algebra

Santino G.

In linear algebra, we are occasionally interested in resolving vectors into their “projections.” In contrast with components, however, which are determined with respect to the x- and y-axis, projections are determined with respect to an adjacent vector, one that shares the same initial point. This blog discusses the derivation of formulas that help us in finding these projections. Suppose we have two nonzero vectors u and v in 2-space. The vector u can be written as a sum of vectors, say, and : (i) where is constructed to be parallel to the adjacent vector v and is made to be … Continue reading Orthogonal projections in linear algebra

Cross products and dot products in linear algebra

Santino G.

This is a short post where I briefly describe cross products and dot products. If you have not previously studied linear algebra, the math may be a tad complicated. Cross products The cross product of two 3D vectors u and v () defines the components of the vector that is orthogonal (perpendicular) to both. That is all! Quite simple. The formula is similarly intelligible. For vectors and , their cross product would be A better way of writing this would be in terms of determinants as so This formula can be remembered by considering the 2×3 matrix whose first row … Continue reading Cross products and dot products in linear algebra

Novel interpretation of the Delta-Epsilon Proof (maybe)

Santino G.

While writing my calculus blog, I found (perhaps even discovered) a new way of thinking about the – (delta-epsilon) proof. This interpretation won’t cause any kind of epistemic revolution in mathematics, but I believe it’s a helpful pedagogical tool worthy of its own blog post. In case you have forgotten, the limit definition states the following (where and are real numbers): To prove a limit statement, we have to show that the limit satisfies this definition. How do we do this? We assume exists and show that, no matter what the value of is, there exists a corresponding . The … Continue reading Novel interpretation of the Delta-Epsilon Proof (maybe)

The complexity of calculus is only superficial

Santino G.

Calculus provided relief from the two-thousand-year decline in mathematics that proceeded the death of Archimedes. With its tools to analyze motion, change, and infinite series, calculus was a novel interpretation of reality and, to many, impossible to understand. Probably more often than not, students have gone into studying this subject with rumors of its difficulty floating in their mind, and, accordingly, they find it intractable. After years of calculus study, however, I find this widespread impression to be false. I cannot generalize my claim over the entirety of calculus just yet, considering I am still only on Calc II, but … Continue reading The complexity of calculus is only superficial