Mathematics
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Adding and subtracting vectors made simple
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Understanding linear transformations intuitively
Matrices as Machines Most people first encounter matrices as lifeless grids, rows, and columns filled with numbers. The rules for working with them seem arbitrary. Multiply this row by that column. Follow this sequence. Don’t ask why. But there’s a far more intuitive, visual way to understand what’s going on. A matrix is a machine.…
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One more time: What is Linear Algebra
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Avoiding Redundancy: Linear Independence
(or: Why packing two hairdryers is a terrible idea) Picture this: you’re packing for a short trip. You double-check your bag, and, just to be safe, you toss in two identical hairdryers. Same size, same model, same everything. Unless you’re secretly a stylist on tour, that second hairdryer? Pointless. It adds weight without adding value.…
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Vectors: The Building Blocks of Linear Algebra
Picture giving directions: “Take 3 steps forward, then 4 to the right.” That’s a vector, a quantity with both direction and magnitude. In math, it’s written like this:v = [3, 4] This notation shows how far to move in each dimension. It’s concise, practical, and powerful. Vectors appear in physics (forces), video games (movement), data…
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Linear Algebra
Welcome to a new set of blog posts that will explore Linear algebra. At its core, linear algebra is a branch of mathematics that deals with: Vectors: These are quantities that have both a magnitude and a direction, like arrows. Vector Spaces: These are sets of vectors that follow certain rules. You can add vectors…
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The Role of Integrals in Solving Differential Equations
Differential equations are at the core of mathematical modeling. They describe how things change. This could be the motion of a pendulum. It might also refer to the growth of a population. Moreover, it could mean the cooling of a metal rod. But identifying the equation is only part of the task. The real challenge…
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Unbounded limits
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Understanding Improper Integrals in Calculus
In calculus, integration is often introduced as a way to measure area under a curve. For most functions and intervals, this is straightforward. But not all mathematical problems are that well-behaved. Some functions spike to infinity. Others stretch endlessly across the number line. Standard integrals can’t handle these cases. Improper integrals can. What Are Improper…
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Integrals: Tools for Measuring Economic Surplus
In economics, the value of a transaction is not measured solely by the price paid. Behind every market exchange lies a surplus, value gained by both consumers and producers that goes beyond the monetary cost. Understanding these surpluses is critical to evaluating the efficiency and fairness of markets. Integrals, although mathematical in origin, serve as…