Solve It - Unit Portfolio
Unit Question: “How can we solve equations, and why does this matter?”
Throughout the history of humanity, we have discovered that many natural scenarios are most accurately represented with mathematical equations and expressions. We have found many different ways to solve these problems, and we're constantly finding more. In math, we covered a multitude of different areas, from changing units, to dividing polynomials.
I myself have been working on linear algebra. More specifically, how to solve linear systems of equations. A linear system of equations is a way to describe the relationships between several unknown values, and how those values behave when they are added or subtracted. One way to represent these systems is with matrix-vector multiplication, and... well... at this point you might be confused, which brings me to a detour, and one of my biggest pet-peeves in mathematics. |
More commonly in middle school, mathematics and mathematical ideas are described and taught as rules and formulas that should be memorized and mastered, which I find to be not only a blissfully alliterative sentence, but one of the biggest flaws of our educational system. Mathematics, (to me at least), is all about the art of problem solving and discovery, not about copy-and-pasting algebra problems from mathisfun.com onto paper, and then handing it to a group of unprepared 13-year-olds.
More often than not, this exposure leads to a particular distaste towards math, which didn't come from the content itself, but came from the way it was taught. So in the extraordinary event that a fellow youngin is actually reading this recreationally, I would like to offer you a nerd's perspective, math is beautiful, and not youtube mandelbrot set thumbnail beautiful, really beautiful. Now, back to linear algebra.
More often than not, this exposure leads to a particular distaste towards math, which didn't come from the content itself, but came from the way it was taught. So in the extraordinary event that a fellow youngin is actually reading this recreationally, I would like to offer you a nerd's perspective, math is beautiful, and not youtube mandelbrot set thumbnail beautiful, really beautiful. Now, back to linear algebra.
First, we need to define a couple things. A vector is an arrow, most commonly an arrow that sits in the 2d plane or some other dimension, and has a direction. You know what? Just look at the picture. A matrix is a function that transforms a vector, squishing it or stretching it, as long as its tail sits at the origin. A linear transformation can be thought of as asking the question, "If we apply this matrix-function to every single vector, which one lands on this vector, v?" This is more commonly notated as Mx = v, where M is a matrix, x is a vector with unknown coordinates, and v is a vector whose coordinates we do know. In order to solve this equation, we need to find the inverse of M, or M⁻¹. M⁻¹ is basically the transformation that "reverses" the M transformation. So, x is defined to be M⁻¹v. We find the inverse with Gaussian Elimination, which is essentialy applying a bunch of operations to the rows of a matrix, untill we get the Identity Matrix, or the matrix that does nothing. Learn more about Linear Algebra here.
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