Scalar Multiplication

Linear Algebra for Machine Learning · Axiom Academy

Scaling vectors: stretching and shrinking What is scalar multiplication? Multiply each component of a vector by a single number (called a scalar). What does scaling do geometrically? c > 1: Vector stretches away from origin (becomes longer) 0 < c < 1: Vector shrinks toward origin (becomes shorter) c < 0: Vector points in opposite direction and scales c = 0: Vector becomes the zero vector [0, 0, ...] Multiply each component by 3. We get [6, 12, 15] — a vector in the same direction but 3 times longer. Multiply each component by -0.5. The negative sign reverses direction, and the 0.5 shrinks the magnitude. The result points opposite and is half as long. Question 1: Understanding Scalar Multiplication If v = [2, -1, 3], what is 4v? Question 2: Geometric Interpretation If we compute -1 times a vector v, what happens geometrically?

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