6/20/2023 0 Comments Khan linear algebra![]() ![]() Consider the “add three” function $F(x) = x + 3$: ![]() Surprisingly, regular addition isn’t linear either. We doubled the input but quadrupled the output. Which operations are linear and predictable? Multiplication, it seems.Įxponents ($F(x) = x^2$) aren’t predictable: $10^2$ is 100, but $20^2$ is 400. In our example, $F(x)$ calculates the rise when moving forward x feet, and the properties hold:Īn operation is a calculation based on some inputs. In math terms, an operation F is linear if scaling inputs scales the output, and adding inputs adds the outputs: If 3 feet forward has a 1-foot rise, and 6 feet has a 2-foot rise, then (3 + 6) feet should have a (1 + 2) foot rise.If 3 feet forward has a 1-foot rise, then going 10x as far should give a 10x rise (30 feet forward is a 10-foot rise).Contrast this with climbing a dome: each horizontal foot forward raises you a different amount. Move forward 6 feet, and you’d expect a rise of 2 feet. Imagine a rooftop: move forward 3 horizontal feet (relative to the ground) and you might rise 1 foot in elevation (The slope! Rise/run = 1/3). ![]() “Linear Algebra” means, roughly, “line-like relationships”. Without knowing x and y, we can still work out that $(x + y)^2 = x^2 + 2xy + y^2$. Grade-school algebra explores the relationship between unknown numbers. “Algebra” means, roughly, “relationships”. Here’s the linear algebra introduction I wish I had, with a real-world stock market example. It’s the power of a spreadsheet written as an equation. We can take a table of data (a matrix) and create updated tables from the original. Linear algebra gives you mini-spreadsheets for your math equations. The survivors are physicists, graphics programmers and other masochists.
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