Here’s how I visualize inputs, operations, and outputs: We’re getting organized: inputs in vertical columns, operations in horizontal rows. We should separate the inputs into groups: 1st Input 2nd InputĪnd how could we run the same input through several operations? Have a row for each operation: F: 3 4 5 We could try this:īut it won’t work: F expects 3 inputs, not 6. Let’s spice it up: how should we handle multiple sets of inputs? Let’s say we want to run operation F on both (a, b, c) and (x, y, z). We know to multiply the first input by the first value, the second input by the second value, the third input by the third value, and add the results. We could abbreviate the entire function as (3, 4, 5). Next, how should we track our operations? Remember, we only have “mini arithmetic”: multiplications by a constant, with a final addition. We could write it (x, y, z) too - hang onto that thought. First, how should we track a bunch of inputs? How about a list: x
We generate a result, perhaps transforming it again. We have predictable, linear operations to perform (our “mini-arithmetic”). Most courses hit you in the face with the details of a matrix. Limiting ourselves to linear operations has its advantages. If we allowed non-linear operations (like $x^2$) we couldn’t split our work and combine the results, since $(a+b)^2 \neq a^2 + b^2$. It’s actually useful because we can split inputs apart, analyze them individually, and combine the results: We have “mini arithmetic”: multiply inputs by a constant, and add the results. $G$ is still linear, since doubling the input continues to double the output: We can still combine multiple linear functions ($A(x) = ax, B(x) = bx, C(x)=cx$) into a larger one, $G$: In our roof example, $a = 1/3$.īut life isn’t too boring. So, what types of functions are actually linear? Plain-old scaling by a constant, or functions that look like: $F(x) = ax$. (Yes, $F(x) = x + 3$ happens to be the equation for an offset line, but it’s still not “linear” because $F(10) \neq 10 \cdot F(1)$. We doubled the input and did not double the output. 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. Favor abstract examples (2d vectors! 3d vectors!) and avoid real-world topics until the final week. Teach concepts like Row/Column order with mnemonics instead of explaining the reasoning. Name the course Linear Algebra but focus on things called matrices and vectors. Despite two linear algebra classes, my knowledge consisted of “Matrices, determinants, eigen something something”.
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