Linear Algebra

From ForcaWiki, the simple encyclopedia

Imagine giving a friend directions to your house using an arrow: "walk 3 blocks east and 2 blocks north." That arrow is really just two numbers, 3 and 2, but it also has a direction and a length. In math, a list of numbers like that is called a vector. Linear algebra is the part of math that studies vectors, and the tools that move, stretch, or spin them around.

A vector does not have to be about walking somewhere. It can be a list of ingredients in a recipe, the red, green, and blue amounts that make up a color on a screen, or the x, y, and z position of a spaceship in a video game. Anywhere you have a group of numbers that belong together and mean something as a set, you probably have a vector.

Grids that transform vectors

Linear algebra also has a second main character: the matrix, which is just a rectangular grid of numbers, arranged in rows and columns. A matrix is like a machine. You feed a vector into it, and it hands back a new vector, changed in some tidy way, such as stretched longer, spun around, or flipped like a reflection in a mirror.

Think about a photo on your phone. Rotating it, zooming in, or flipping it upside down can all be done by feeding every point of the picture, each one really a small vector, through the right matrix. The matrix holds the instructions for exactly how to twist, stretch, or turn everything.

Solving many clues at once

Linear algebra is also the math you use when you have several clues about some numbers and need to find the numbers that fit all the clues at once. For example, if you know that two toys together cost 10 dollars, and one toy costs 2 dollars more than the other, linear algebra gives you an organized way to line up both clues and solve for both prices together, rather than guessing and checking.

Where you find it

Linear algebra quietly powers an enormous amount of modern technology:

• Video games and movies use it to move, rotate, and light up 3D characters and scenes.
• Photo and video filters use it to shift colors and warp images.
• Map apps use it to figure out routes and distances.
• The computer programs behind translation, recommendations, and image recognition are built out of enormous vectors and matrices multiplied together, which is part of why understanding Probability alongside linear algebra matters so much in modern computing.

Fun facts

• The word "vector" comes from a Latin word meaning "carrier," since a vector carries both a size and a direction, the way an arrow carries you a certain distance in a certain direction.
• Long before computers existed, linear algebra was already being used to plot the paths of planets and to design bridges.
• A single, simple matrix can rotate an entire 3D character in a video game hundreds of times every second, fast enough that it looks like smooth, live motion to your eyes.