A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A linear transformation T: Rⁿ → Rᵐ satisfies T(cu + v) = cT(u) + T(v) for all scalars c and vectors u, v. Linear transformations preserve vector addition and scalar multiplication, making them algebraic homomorphisms. Every linear transformation is represented by a unique matrix A such that T(x) = Ax.
You already know functions from earlier math — a function takes an input and produces an output. A linear transformation is a special kind of function that takes *vectors* as inputs and produces *vectors* as outputs, subject to two constraints that make it structurally well-behaved. These constraints are what give linear algebra its power.
The two conditions are: T(u + v) = T(u) + T(v), and T(cv) = cT(v). Together they can be compressed into the single condition T(cu + v) = cT(u) + T(v). Intuitively, this says it doesn't matter whether you "do the algebra first, then transform" or "transform first, then do the algebra" — you get the same answer. A transformation with this property is one we can analyze, compose, and invert in a clean, predictable way.
A critical consequence: every linear transformation sends the zero vector to the zero vector. Proof: T(0) = T(0·v) = 0·T(v) = 0. This gives you a quick test — if a function sends any input to a nonzero output when all inputs are zero, it is not linear. This disqualifies functions like T(x) = x + 1, which look almost linear but fail the zero-vector test.
The connection to matrices is fundamental: every linear transformation T: ℝⁿ → ℝᵐ can be represented by an m×n matrix A, where T(x) = Ax. To find A, you only need to know what T does to the standard basis vectors — linearity then determines T's behavior everywhere else. This matrix representation is the bridge to eigenvalues, determinants, and the rest of linear algebra.
Geometrically, linear transformations on ℝ² and ℝ³ include rotations, reflections, scaling, and projections — all operations that map straight lines to straight lines and keep the origin fixed. Non-linear operations like "shift everything right by 1" fail to be linear precisely because they move the origin. Keeping this geometric picture in mind helps you check whether a given transformation can possibly be linear.