Vector Addition & Subtraction

Linear Algebra for Machine Learning · Axiom Academy

Combining vectors: geometry and algebra Vector addition is simple: add corresponding components. Each component of the result is the sum of the corresponding components. Example: If u = [2, 3] and v = [1, 4], then u + v = [2+1, 3+4] = [3, 7] Vector addition has a beautiful geometric meaning: the tip-to-tail rule . To add vectors geometrically: place the tail of the second vector at the tip of the first. The sum vector goes from the origin to the final tip. Order doesn't matter. Adding geometrically in either order gives the same result. Grouping doesn't matter. We can add three vectors in any grouping order. Adding the zero vector [0, 0, ...] doesn't change anything. Every vector has an opposite that sums with it to give zero. Why this matters: These properties make vectors a fundamental structure in mathematics called a "group under addition." This structure appears everywhere in science and engineering!

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