Linear Combinations

Linear Algebra (Matrices) · Axiom Academy

LESSON Linear Combinations & Span The exact recipe for building one vector out of others — a weighted sum — and the name for everything that recipe can reach. 1. A Linear Combination Is a Weighted Sum Take vectors and pick two numbers — the weights (or scalars ) c_1,c_2 . Scale each vector by its weight and add the pieces tip-to-tail. The vector you land on is a linear combination of and . Watch the weights assemble the result. scale each vector by its weight, then add 2. Different Weights Build Different Vectors A linear combination isn't one fixed vector — it's a recipe with adjustable weights. Every choice of (c_1,c_2) produces its own result vector. The general form, for any number of vectors, is a single weighted sum. a linear combination of with scalar weights c_i Each labelled dot above is one combination . Same two vectors, different weights, different landing point: Only contributes — you slide out along its direction to (6,2) . Only contributes — landing at (2,4) . One of each: (3,1)+(1,2)=(4,3) . A negative weight flips before adding: (-3,-1)+(2,4)=(-1,3) . 3. The Span Is Everything You Can Reach Now let the weights c_1,c_2 run over all real numbers at once. The set of every linear combination you can possibly build is called the span of . Watch the landing tips sweep out and fill the region they cover. Here, the span is the whole plane

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