Definition and Properties

Linear Algebra (Matrices) · Axiom Academy

LESSON Defining a Linear Transformation A map is linear exactly when it preserves the two vector-space operations — addition and scalar multiplication. 1. The Definition, and the First Axiom A function between vector spaces is a linear transformation when it satisfies two conditions for all vectors and every scalar c : Additivity — T commutes with addition Homogeneity — T commutes with scaling Read the first axiom geometrically. The vectors and span a parallelogram whose diagonal is the sum . Additivity says: apply T to that whole picture, and the image is still a parallelogram whose diagonal closes — so lands exactly on . 2. The Second Axiom: Scaling Passes Through The second axiom is about length rather than tip-to-tail addition. Stretch the input by a factor c before applying T , or apply T first and stretch the output by c — homogeneity says you get the same vector either way. In the animation, watch the two arrows grow together: as the input scales up to , its image scales by the same factor c , from out to . The scalar slides straight through the transformation. 3. What the Axioms Force — and What Breaks Them Two facts fall out immediately, before you compute anything. Setting c=0 in homogeneity gives : a linear map must fix the origin . And applying superposition repeatedly extends it to every linear combination, so .

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